8
$\begingroup$

I want to find $$f(n) = \max\left\{\frac{\varphi(k)}{\lambda(k)} : 1 \leq k \leq n\right\}$$

In other words, I want to find the maximal value of $\frac{\varphi(k)}{\lambda(k)}$ when $k$ is restricted.

$\lambda(n)$ is the Carmichael function, the smallest $k$ such that for all $a$ such that $\gcd(a,n)=1$ the following congruence holds: $a^{k} \equiv 1 \mod n$. $\varphi(n)$ is the Euler-phi function. Note that $\lambda(n) \mid \varphi(n)$.

Since I don't expect $f(n)$ to have a nice expression in terms of elementary and number theoretic functions, I started to search for some lower bounds. I found that $$\frac{\varphi(2^k \cdot p_{k+1}\#)}{\lambda(2^k \cdot p_{k+1}\#)} \geq 2^k$$

Here $k\#$ is the primorial function.

Therefore I know that the fraction is unbounded, and that $$f(n) \in \Omega(n^{1/ \ln \ln n})$$

However $n^{1/ \ln \ln n}$ is still bounded above by all power functions with a positive power. Does anyone have an asymptotically better bound or a proof that this is the best bound?

I looked to OEIS, which has this sequence A034380. I found that the smallest $n$ such that $f(n)\geq 108$ is $n=3591$, the smallest $n$ such that $f(n)\geq 128$ is $n=5440$. However my method gives $1241560320$ as a number such that it is bigger than 128. Clearly, there is some room for improvement.

Conjecture: Will Jagy found strong numerical evidence that $f(n)$ eventually outgrows $n^{1-\varepsilon}$ for every $\varepsilon > 0$. A proof (or disproof) of this would be very welcome.

$\endgroup$
  • 1
    $\begingroup$ I haven't yet thought about formalising it, but if you take a [sufficiently] highly composite number $n$ and start from a prime $p = kn+1$ with small $k$, you can throw in a lot of smaller prime powers that don't increase $\lambda$ (much) but of course increase $\varphi$. Starting from $n = 24$, I took $p = 73$. Then we can take small prime powers and obtain $$m = 2^4\cdot 3^3 \cdot 5\cdot 7 \cdot 13\cdot 17 \cdot 19 \cdot 37 \cdot 73 = 171483464880$$ with $\lambda(m) = 144$ and $\varphi(m) = 30958682112 = 214990848\cdot\lambda(m)$. $\endgroup$ – Daniel Fischer Jun 22 '15 at 11:05
  • 1
    $\begingroup$ If we start from $19 = 3\cdot 6 + 1$, we can throw in $2^3, 3^3,7$ without increasing $\lambda$ at all - lo and behold, $3^3\cdot 7 \cdot 19 = 3591$. $\endgroup$ – Daniel Fischer Jun 22 '15 at 11:09
  • $\begingroup$ @DanielFischer In other words, does this mean that it is $O(n)$? $\endgroup$ – wythagoras Jun 22 '15 at 11:30
  • $\begingroup$ One motivation for this question is that $\varphi(n)$ is the order of $U(n)$ and $\lambda(n)$ is the exponent. Here $U(n)$ is the group of units mod $n$. $\endgroup$ – lhf Jun 22 '15 at 11:35
  • $\begingroup$ I suspect that $f \in o(n)$, but I don't think $\frac{f(n)}{n}$ converges to $0$ fast. I don't make a guess about the asymptotics yet, however. $\endgroup$ – Daniel Fischer Jun 22 '15 at 12:00
6
+50
$\begingroup$

This time I let the target Carmichael number be the least common multiple of the numbers from $1$ to $w.$ This is more efficient in terms of the number of divisors. The Superior Highly Composite Numbers and the Colossally Abundant Numbers share the main property of this LCM, which is that the exponent of some prime $p$ is proportional to $1/ \log p.$ As a result, the multiplicative contribution from each prime is (very) roughly equal, because $$ x^{1/ \log x} = e. $$

We know from Legendre's theorem on the exponent of a prime dividing a factorial that the exponent here of $p$ is proportional to $1/p.$ So the smaller primes give a large multiplicative part, as $x^{1/x}$ goes to $1$ as $x \rightarrow \infty.$ Not an astonishing difference, I guess.

The program got incredibly slow doing the lcm of the numbers up to $23,$ so I will just paste in the lcm's up to $19.$

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12
 carm  12 = 2^2 3
 n 65520
 n  = 2^4 3^2 5 7 13
  log n  11.0901
Euler Phi 13824
Euler Phi  = 2^9 3^3
  Euler Phi / Carmichael 1152
  Euler Phi / Carmichael  = 2^7 3^2
 log ( Euler Phi / Carmichael)   7.04925
 log ( Euler Phi / Carmichael) / log n  0.635634
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60
 carm  60 = 2^2 3 5
 n 6814407600
 n  = 2^4 3^2 5^2 7 11 13 31 61
  log n  22.6423
Euler Phi 1244160000
Euler Phi  = 2^13 3^5 5^4
  Euler Phi / Carmichael 20736000
  Euler Phi / Carmichael  = 2^11 3^4 5^3
 log ( Euler Phi / Carmichael)   16.8474
 log ( Euler Phi / Carmichael) / log n  0.744067
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420
 carm  420 = 2^2 3 5 7
 n 375159113055066740400
 n  = 2^4 3^2 5^2 7^2 11 13 29 31 43 61 71 211 421
  log n  47.3739
Euler Phi 63233645690880000000
Euler Phi  = 2^20 3^8 5^7 7^6
  Euler Phi / Carmichael 150556299264000000
  Euler Phi / Carmichael  = 2^18 3^7 5^6 7^5
 log ( Euler Phi / Carmichael)   39.5531
 log ( Euler Phi / Carmichael) / log n  0.834914
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840
 carm  840 = 2^3 3 5 7
 n 8644416283014847832296800
 n  = 2^5 3^2 5^2 7^2 11 13 29 31 41 43 61 71 211 281 421
  log n  57.419
Euler Phi 1416433663475712000000000
Euler Phi  = 2^27 3^8 5^9 7^7
  Euler Phi / Carmichael 1686230551756800000000
  Euler Phi / Carmichael  = 2^24 3^7 5^8 7^6
 log ( Euler Phi / Carmichael)   48.8768
 log ( Euler Phi / Carmichael) / log n  0.851231
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2520
 carm  2520 = 2^3 3^2 5 7
 n 48665323350093056511370687590824766511200
 n  = 2^5 3^3 5^2 7^2 11 13 19 29 31 37 41 43 61 71 73 127 181 211 281 421 631 2521
  log n  93.6858
Euler Phi 7138535724796543865494437888000000000000
Euler Phi  = 2^40 3^23 5^12 7^10
  Euler Phi / Carmichael 2832752271744660264085094400000000000
  Euler Phi / Carmichael  = 2^37 3^21 5^11 7^9
 log ( Euler Phi / Carmichael)   83.9343
 log ( Euler Phi / Carmichael) / log n  0.895913
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27720
 carm  27720 = 2^3 3^2 5 7 11
 n 46835535664453140916928878375591275233836162863207874618151609888175819169346400
 n  = 2^5 3^3 5^2 7^2 11^2 13 19 23 29 31 37 41 43 61 67 71 73 89 127 181 199 211 281 331 397 421 463 617 631 661 991 1321 2311 2521 4621 9241
  log n  183.448
Euler Phi 6283868711272246950118489040817069058881601596869138196725760000000000000000000
Euler Phi  = 2^65 3^37 5^19 7^15 11^15
  Euler Phi / Carmichael 226690790449936758662283154430630196929350706957761118208000000000000000000
  Euler Phi / Carmichael  = 2^62 3^35 5^18 7^14 11^14
 log ( Euler Phi / Carmichael)   171.21
 log ( Euler Phi / Carmichael) / log n  0.933286
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360360
 carm  360360 = 2^3 3^2 5 7 11 13
 n 298871636051787972324640321489709180758102154467075853983420470349998567281384922961858797094684848792536021545048660618505111461843580370418446603390641898939152029204827347934755791200
 n  = 2^5 3^3 5^2 7^2 11^2 13^2 19 23 29 31 37 41 43 53 61 67 71 73 79 89 127 131 157 181 199 211 281 313 331 397 421 463 521 547 617 631 661 859 911 937 991 1093 1171 1321 2003 2311 2341 2521 2731 2861 3433 4621 6007 6553 8009 8191 8581 9241 16381 20021 25741 36037 51481 72073 120121 180181
  log n  427.073
Euler Phi 37736396954157598059788348310956484406049704954111539794809548644964811972308825522143018925098662022578445724996281865982823373310042400379419838382080000000000000000000000000000000000
Euler Phi  = 2^124 3^70 5^34 7^30 11^29 13^31
  Euler Phi / Carmichael 104718606266393601009513676076580320807108738356397879328475825965603318826475817299764177281326068438723625610490292668395003255938623599676489728000000000000000000000000000000000
  Euler Phi / Carmichael  = 2^121 3^68 5^33 7^29 11^28 13^30
 log ( Euler Phi / Carmichael)   412.209
 log ( Euler Phi / Carmichael) / log n  0.965195
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720720
 carm  720720 = 2^4 3^2 5 7 11 13
 n 2599265289938045790285087430718636500803510968991503583932015415681339876886274198699629634541272594255929473856752109528189646768583309952911477951292254958722748988899531120282182905307875781283119196897221941548215066799861837732382400
 n  = 2^6 3^3 5^2 7^2 11^2 13^2 17 19 23 29 31 37 41 43 53 61 67 71 73 79 89 113 127 131 157 181 199 211 241 281 313 331 337 397 421 463 521 547 617 631 661 859 881 911 937 991 1009 1093 1171 1321 1873 2003 2311 2341 2521 2731 2861 3121 3433 3697 4621 6007 6553 8009 8191 8581 9241 16381 18481 20021 20593 21841 25741 36037 48049 51481 55441 65521 72073 120121 180181
  log n  546.668
Euler Phi 302928484859039757167917387259074936375908184912676644232090483303653116305985172517367761646862734195251724208088514016554177810120723602606503842865373298717800011581324246392169423151903329484800000000000000000000000000000000000000000
Euler Phi  = 2^185 3^87 5^41 7^39 11^35 13^37
  Euler Phi / Carmichael 420313693055610718681204056026022500244072850639189483061508607092425791300345727213574982860004903700815468154190967388936310647853151851768375850351555803526751042820130211999347073970339840000000000000000000000000000000000000000
  Euler Phi / Carmichael  = 2^181 3^85 5^40 7^38 11^34 13^36
 log ( Euler Phi / Carmichael)   531.03
 log ( Euler Phi / Carmichael) / log n  0.971395
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12252240
 carm  12252240 = 2^4 3^2 5 7 11 13 17
 n 7840740066841325159663012343109406035896969282188089816022023253073930685593138754608115114178718941653356910879846676628187748417141835552934771418947605257522367259619199969384442710324874085579531050831286424944502736136712129141705080899663945058655290616070228265094769144920132517587558228976893437743186457341466720980209808301629382042408853888361938252926075102952852024516523051299362023013137353239343091598865047297527610474958115324782865413736021297404821213490729251550378837210338248063398056065100020800
 n  = 2^6 3^3 5^2 7^2 11^2 13^2 17^2 19 23 29 31 37 41 43 53 61 67 71 73 79 89 103 113 127 131 137 157 181 199 211 239 241 281 307 313 331 337 397 409 421 443 463 521 547 613 617 631 661 859 881 911 937 953 991 1009 1021 1093 1123 1171 1321 1327 1361 1429 1531 1871 1873 2003 2143 2311 2341 2381 2521 2731 2857 2861 3061 3121 3433 3571 3697 4421 4621 5237 6007 6121 6553 6733 7481 8009 8191 8581 9241 9283 9521 10711 12241 12377 14281 15913 16381 16831 17137 17681 18481 19891 20021 20593 21841 22441 23563 25741 27847 29173 30941 36037 42841 43759 46411 48049 51481 52361 55441 65521 72073 72931 74257 78541 79561 87517 92821 97241 102103 116689 117811 120121 145861 157081 180181 185641 235621 291721 314161 371281 471241 612613 680681 816817 4084081
  log n  1197.1
Euler Phi 876963786249831215329065464481713945416735673133399063326739418167466275084490195155585298354804980317157310709931920654014082320754625938193488917188172905629711630411642807892170926210559770331095200711744675075568998096103910242973547372095768936891066690134738650385202960878682027534352001136594406411167332321078588661428170071187615269703076016409856809942317092056015529093210439558720962836997515774679821891872505495454221714063360000000000000000000000000000000000000000000000000000000000000000000000000000000
Euler Phi  = 2^339 3^160 5^79 7^74 11^63 13^66 17^68
  Euler Phi / Carmichael 71575792365300648316476453651064127491522829550629033003494823654080092708312128652033040354645761127529113917939243816152318459380050173535083292294974054183538000431891866947772074837789642574018726429758531915434973367817142844326714737231377196079334610661784183984740991106824713483767213271744138737991365849924470028454239393873088942895591011636227890568770860843079757586629909270363701889368598376678862142095853941438808064000000000000000000000000000000000000000000000000000000000000000000000000000000
  Euler Phi / Carmichael  = 2^335 3^158 5^78 7^73 11^62 13^65 17^67
 log ( Euler Phi / Carmichael)   1178.59
 log ( Euler Phi / Carmichael) / log n  0.984536
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232792560
 carm  232792560 = 2^4 3^2 5 7 11 13 17 19
 n 50534333093172223543129940291654815617151944146476851368660955499649474633663170820397745391407946393810344018943466673835027481077690065639852829994132892247661970216297524131089684486076569988963974748490096400953601986443230692489382127281828207476930926656993032159692158359213103542287271323853769097052208533488613670497659058052282054082188507391900881880619583306518097921009261508681154148168035945026167174774487564624431388560148364227110384299217431751777238418503876422038688191658095555268373577241322157898647844988230539117410212547101213436222907309137876912437504569908503281850569541099592401796096523635543039626980406691563396551444451745862370202667382601059271275160260669005385300437576767917007146159132580605728714248593018133559184952841625792695276539620965459288444034412706624418909888049690529619372510185042850423332654357883876922840520857879178067105509474843367530324617706324050546948962356054939237309734274832559183894772446907064920899799767179897019983498925857042697497845396145415103015404452800
 n  = 2^6 3^3 5^2 7^2 11^2 13^2 17^2 19^2 23 29 31 37 41 43 53 61 67 71 73 79 89 103 113 127 131 137 157 181 191 199 211 229 239 241 281 307 313 331 337 397 409 419 421 443 457 463 521 547 571 613 617 631 647 661 761 859 881 911 937 953 991 1009 1021 1093 1123 1171 1321 1327 1361 1429 1483 1531 1597 1871 1873 2003 2129 2143 2281 2311 2341 2381 2521 2731 2857 2861 2927 3061 3121 3433 3571 3697 3877 4421 4447 4523 4561 4621 4789 5237 6007 6121 6271 6553 6733 6841 6917 7411 7481 7753 8009 8191 8581 8779 8893 9241 9283 9521 10711 11971 12241 12377 12541 13567 13681 14281 14821 15913 16381 16831 17137 17291 17681 18089 18481 19381 19891 20021 20593 20749 21319 21737 21841 22441 23563 25741 25841 27847 29173 29641 30097 30941 31123 35531 35569 36037 38039 40699 42841 43759 43891 46411 47881 48049 48907 51481 51871 52361 55441 58787 59281 65521 72073 72931 74257 75583 77521 78541 79561 87517 92821 97241 97813 102103 105337 106591 108529 116689 117041 117811 120121 124489 131671 135661 145861 157081 163021 177841 180181 185641 207481 213181 217361 235621 251941 291721 302329 314161 342343 351121 371281 377911 391249 406981 456457 461891 471241 489061 511633 526681 554269 612613 652081 680681 813961 816817 895357 1053361 1058149 1108537 1279081 1369369 1492261 1790713 1939939 2217073 2238391 2282281 2351441 2645371 2771341 2984521 3233231 4084081 4157011 4232593 4476781 6651217 6846841 17907121 46558513 232792561
  log n  2387.1
Euler Phi 5514449822311644815404751686462977951034810669560490390518583345450784157372094367894048911784096083376344400979305654850358459153953986987336008744921030392692447550713150394082012450128757329333903497505348213207876674433801907491791066625979405161798227640380343910277367189022142510413519613475494507166995486198422310655527868738996899023727607093793576815536756577970675943000561603741541848237145541145404230173466423531210533394354489728605628509391301279454516247049568108488644469334042560684856963327508699823730819658908650485763190516837544747349671010096018387743869026349184431183582691118568072484260050981261437067547241103600317134737994971149802615231089264186114925784384959460111359636215747257380703663933037572553721728696626117704554050895418557994555303296028367133022206932589927112399034402488153862223577676893982895775184698996047515779443891083701369171975300626668270387200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
Euler Phi  = 2^584 3^287 5^134 7^125 11^114 13^116 17^115 19^105
  Euler Phi / Carmichael 23688256284099649986257085219832532238293228398538554627856591917932360713641769169487413651811278175627023479527462797137324574092720089453614878177038950010655183957396019847378337392435382511081554743439172683215806701184101018914827289265513490473227441806475017544707473421926123886491559753780337770103114490421954682123551838336229040239634836670869450533714464835004503335504199978476725580221058358331573097411130422429353126209679938777277196957631727059724401188120308090982995630676695856108360865688786187254999986506908341425358226726994817821281191332300389616162428156420396043514374733963010125771459581789303906738029948652999550908061644973317887028825531469674610416176466118419383160854521069132882527104530478003909238889321145476919683562461869734988761252919888707495730133869355305480549010683537969865633066954089868231936556301438703693019415616563095354817075342213120000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
  Euler Phi / Carmichael  = 2^580 3^285 5^133 7^124 11^113 13^115 17^114 19^104
 log ( Euler Phi / Carmichael)   2365.62
 log ( Euler Phi / Carmichael) / log n  0.991001
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Got a much abbreviated output, so the whole thing goes much faster and I am able to compare things. It turns out the first time $\log f(n) / \log n$ passes 0.999 is actually when the assigned Carmichael number is the LCM up to $31,$ and the first time we pass 0.9999 is when the assigned Carmichael number is the LCM up to $43.$ I had it put in logs base 10 as well.

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      log  n 45666.52180017586  base  ten  19832.71842553093
     euler_log 45664.14300016323  base  ten  19831.68532581189
     carm  72201776446800 = 2^4 3^3 5^2 7 11 13 17 19 23 29 31
     carm log 31.91048576605096  base  ten  13.85854788304819
     ratio so far  0.9992491373455432
     carm final 72201776446800 = 2^4 3^3 5^2 7 11 13 17 19 23 29 31
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      log  n 497887.2342748803  base  ten  216229.6784556521
     euler_log 497884.7681825354  base  ten  216228.6074453548
     carm  9419588158802421600 = 2^5 3^3 5^2 7 11 13 17 19 23 29 31 37 41 43
     carm log 43.689323041653  base  ten  18.97403191507849
     ratio so far  0.9999072974516936
     carm final 9419588158802421600 = 2^5 3^3 5^2 7 11 13 17 19 23 29 31 37 41 43
    =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  log  n 6298975.807663561  base  ten  2735610.434910363
 euler_log 6298973.277771472  base  ten  2735609.336192189
 carm  9690712164777231700912800 2^5 3^3 5^2 7^2 11 13 17 19 23 29 31 37 41 43 47 53 59
 carm log 57.53321014987621  base  ten  24.9863556942714
 ratio so far  0.9999904646240797
 carm final 9690712164777231700912800 2^5 3^3 5^2 7^2 11 13 17 19 23 29 31 37 41 43 47 53 59
 lcm of up to:  59 divisors : 1769472  prime count:  221800 log ratio: 3.018864087268213
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

As far as proving anything, let me describe the construction this way: for a number $L,$ we let $C = \operatorname{lcm} (1,2,3,\ldots,L).$ We find all positive divisors $d | C.$ For each such $d,$ we check $q = 1 + d.$ If $q$ is prime, then $q$ becomes a factor of $n.$ Now if, in addition, $q \leq L,$ then $q$ gets an exponent larger than $1,$ because we get $q | C$ this way. In fact, we give $q$ the largest possible exponent such that the Carmichael number of $q^a$ divides $C.$ If, instead, $q > L,$ then the exponent of $q$ is precisely $1.$

This means that the number of primes that arise as $q=1+d$ with $d | L$ is a big part of the giant size of $n.$ It is easy to predict the number of positive divisors of $C$ because we know how it factors. Add one to each, many become prime, many do not. I made a table, it suggests that the number of primes is faster than polynomial as a function of $L,$ but slower than exponential.

lcm of up to:  4 divisors : 6  prime count:  5 log ratio: 1.160964047443681
 lcm of up to:  5 divisors : 12  prime count:  8 log ratio: 1.292029674220179
 lcm of up to:  7 divisors : 24  prime count:  13 log ratio: 1.318123223061841
 lcm of up to:  8 divisors : 32  prime count:  15 log ratio: 1.30229686520284
 lcm of up to:  9 divisors : 48  prime count:  22 log ratio: 1.406794046107798
 lcm of up to:  11 divisors : 96  prime count:  36 log ratio: 1.494443472618428
 lcm of up to:  13 divisors : 192  prime count:  66 log ratio: 1.633425911446773
 lcm of up to:  16 divisors : 240  prime count:  81 log ratio: 1.584962500721156
 lcm of up to:  17 divisors : 480  prime count:  148 log ratio: 1.763796674277192
 lcm of up to:  19 divisors : 960  prime count:  252 log ratio: 1.877922798412615
 lcm of up to:  23 divisors : 1920  prime count:  446 log ratio: 1.945568555358454
 lcm of up to:  25 divisors : 2880  prime count:  660 log ratio: 2.016927706518875
 lcm of up to:  27 divisors : 3840  prime count:  905 log ratio: 2.065616479359747
 lcm of up to:  29 divisors : 7680  prime count:  1638 log ratio: 2.197974766132367
 lcm of up to:  31 divisors : 15360  prime count:  2912 log ratio: 2.322837836698491
 lcm of up to:  32 divisors : 18432  prime count:  3578 log ratio: 2.360987534410382
 lcm of up to:  37 divisors : 36864  prime count:  6661 log ratio: 2.438168109993427
 lcm of up to:  41 divisors : 73728  prime count:  12344 log ratio: 2.536890418395409
 lcm of up to:  43 divisors : 147456  prime count:  23060 log ratio: 2.670917389347842
 lcm of up to:  47 divisors : 294912  prime count:  42735 log ratio: 2.769445392395919
 lcm of up to:  49 divisors : 442368  prime count:  63329 log ratio: 2.840855381850106

Let's see. In the table above, the value of $L$ is on the left, let $P$ be the count of primes constructed. The final number, called "log ratio," is just $\log P / \log L.$ This seems to keep growing, so $P$ seems to be growing faster than a polynomial in $L.$

Next I repeated the table, but this time the "log ratio" is $\log P / L,$ which seems to be decreasing to $0,$ so it seems $P$ is growing slower than an exponential in $L$

 lcm of up to:  4 divisors : 6  prime count:  5 log ratio: 0.4023594781085251
 lcm of up to:  5 divisors : 12  prime count:  8 log ratio: 0.4158883083359671
 lcm of up to:  7 divisors : 24  prime count:  13 log ratio: 0.3664213367802195
 lcm of up to:  8 divisors : 32  prime count:  15 log ratio: 0.3385062751377763
 lcm of up to:  9 divisors : 48  prime count:  22 log ratio: 0.3434491614842574
 lcm of up to:  11 divisors : 96  prime count:  36 log ratio: 0.3257744489505555
 lcm of up to:  13 divisors : 192  prime count:  66 log ratio: 0.3222811340020327
 lcm of up to:  16 divisors : 240  prime count:  81 log ratio: 0.2746530721670274
 lcm of up to:  17 divisors : 480  prime count:  148 log ratio: 0.293953663162595
 lcm of up to:  19 divisors : 960  prime count:  252 log ratio: 0.2910225835532328
 lcm of up to:  23 divisors : 1920  prime count:  446 log ratio: 0.265231258783481
 lcm of up to:  25 divisors : 2880  prime count:  660 log ratio: 0.2596895934008188
 lcm of up to:  27 divisors : 3840  prime count:  905 log ratio: 0.2521457386555528
 lcm of up to:  29 divisors : 7680  prime count:  1638 log ratio: 0.2552148711866557
 lcm of up to:  31 divisors : 15360  prime count:  2912 log ratio: 0.2573095293327928
 lcm of up to:  32 divisors : 18432  prime count:  3578 log ratio: 0.2557049770021458
 lcm of up to:  37 divisors : 36864  prime count:  6661 log ratio: 0.23794661898414
 lcm of up to:  41 divisors : 73728  prime count:  12344 log ratio: 0.2297786681473901
 lcm of up to:  43 divisors : 147456  prime count:  23060 log ratio: 0.2336245300889082
 lcm of up to:  47 divisors : 294912  prime count:  42735 log ratio: 0.2268675220340451
 lcm of up to:  49 divisors : 442368  prime count:  63329 log ratio: 0.225634666103695

There are infinitely many growth rates between polynomial and exponential as a function of $L;$ one of the easiest to type is $$ e^{\sqrt L} $$

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  • $\begingroup$ Thank you for all your hard work. In three hours form now I can start a bounty on this, which will ask for a proof, but if that doesn't come, I will definitely give it to you, furthermore, I hope you will get some more upvotes because of the attention through the bounty. $\endgroup$ – wythagoras Jun 24 '15 at 6:56
  • $\begingroup$ @wythagoras, there might be some hope of showing that the log ratio increases slightly when the lcm increases by one step (one prime exponent increases by one). $\endgroup$ – Will Jagy Jun 24 '15 at 17:14
  • $\begingroup$ The bounty will be yours, but I'll wait awarding it because I hope to get some more upvotes. $\endgroup$ – wythagoras Jun 28 '15 at 11:05
4
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Got a reference on MO; see https://mathoverflow.net/questions/210144/number-of-primes-one-larger-than-divisors-of-a-fixed-number-which-is-lcm-of-1-2#comment520981_210144 and

http://www.math.drexel.edu/~eschmutz/PAPERS/lambda.pdf

In Theorem $1$ on the first page of the Erdos-Pomerance-Schmutz article, they announce the existence of a constant $c$ and a sequence of numbers $n$ going to $\infty,$ such that $$ \lambda(n) < \left( \log n \right)^{c \log \log \log n}. $$

On the other hand, for all $n \geq 3, $ we find in Rosser and Schoenfeld (1962) that $$ \varphi(n) > \frac{n}{ e^\gamma \log \log n + \frac{3}{\log \log n}}. $$

So, on the special sequence in Erdos-Pomerance-Schmutz, we find $$\frac{ \varphi(n)}{\lambda(n)} > \frac{n}{ \left( e^\gamma \log \log n + \frac{3}{\log \log n} \right) \left( \left( \log n \right)^{c \log \log \log n} \right) } $$ which is eventually bigger than any $n^{1-\delta}.$ I am not at all clear what their sequence is.

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2
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It took a while, but I wrote something in C++ with GMP to find the biggest possible $n$ that has a fixed Carmichael number, furthermore I took the Carmichael numbers to be $w!$ for $4 \leq w \leq 12,$ which takes a fair amount of time as it is. The quantity $\log f(n) / \log n$ evidently gets arbitrarily close to $1$ this way.

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24
 carm  24 = 2^3 3
 n 131040 = 2^5 3^2 5 7 13
  log n  11.7833
Euler Phi 27648 = 2^10 3^3
  Euler Phi / Carmichael 1152 = 2^7 3^2
 log ( Euler Phi / Carmichael)   7.04925
 log ( Euler Phi / Carmichael) / log n  0.598243
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120
 carm  120 = 2^3 3 5
 n 558781423200 = 2^5 3^2 5^2 7 11 13 31 41 61
  log n  27.049
Euler Phi 99532800000 = 2^17 3^5 5^5
  Euler Phi / Carmichael 829440000 = 2^14 3^4 5^4
 log ( Euler Phi / Carmichael)   20.5363
 log ( Euler Phi / Carmichael) / log n  0.759224
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720
 carm  720 = 2^4 3^2 5
 n 127589793288205521873600 = 2^6 3^3 5^2 7 11 13 17 19 31 37 41 61 73 181 241
  log n  53.2031
Euler Phi 19258776968232960000000 = 2^34 3^15 5^7
  Euler Phi / Carmichael 26748301344768000000 = 2^30 3^13 5^6
 log ( Euler Phi / Carmichael)   44.733
 log ( Euler Phi / Carmichael) / log n  0.840797
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5040
 carm  5040 = 2^4 3^2 5 7
 n 15321986788854443284662612735663611380010431225771200 = 2^6 3^3 5^2 7^2 11 13 17 19 29 31 37 41 43 61 71 73 113 127 181 211 241 281 337 421 631 1009 2521
  log n  120.161
Euler Phi 2079640131123516572592053607122117591040000000000000 = 2^61 3^27 5^13 7^13
  Euler Phi / Carmichael 412627010143554875514296350619467776000000000000 = 2^57 3^25 5^12 7^12
 log ( Euler Phi / Carmichael)   109.639
 log ( Euler Phi / Carmichael) / log n  0.912432
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40320
 carm  40320 = 2^7 3^2 5 7
 n 6545494715038904244469210654225124710158384371861156116318555122393750667344104768288996697600 = 2^9 3^3 5^2 7^2 11 13 17 19 29 31 37 41 43 61 71 73 97 113 127 181 193 211 241 281 337 421 449 577 631 641 673 1009 1153 2017 2521 2689 3361 4481 13441 20161
  log n  216.019
Euler Phi 866522296042628144676558597700465274142562952474636667572949245734705692672000000000000000000 = 2^143 3^41 5^18 7^21
  Euler Phi / Carmichael 21491128374073118667573377919158364934091343067327298302900526927944089600000000000000000 = 2^136 3^39 5^17 7^20
 log ( Euler Phi / Carmichael)   203.393
 log ( Euler Phi / Carmichael) / log n  0.941548
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362880
 carm  362880 = 2^7 3^4 5 7
 n 114230413363308042027168342906057688115969665979830098456000065958919347571357901455652593685339758276942442955973199255835097303580812111381774457212151783623809025582425600 = 2^9 3^5 5^2 7^2 11 13 17 19 29 31 37 41 43 61 71 73 97 109 113 127 163 181 193 211 241 271 281 337 379 421 433 449 541 577 631 641 673 757 811 1009 1153 1297 1621 2017 2161 2269 2521 2593 2689 3361 3457 4481 6481 7561 8641 10369 12097 13441 15121 20161 30241 45361 72577
  log n  398.48
Euler Phi 14644461873559089811410320023660018881625044215724529464436646866712529435463188980493435700248769685139272910496142474850992293193106871182950400000000000000000000000000000 = 2^227 3^122 5^29 7^30
  Euler Phi / Carmichael 40356211071315833915923500947034884484195999271727649538240318746452076266157377040601399085782544326331770586684695973465036081330210734080000000000000000000000000000 = 2^220 3^118 5^28 7^29
 log ( Euler Phi / Carmichael)   383.624
 log ( Euler Phi / Carmichael) / log n  0.962718
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3628800
 carm  3628800 = 2^8 3^4 5^2 7
 n 68790131970419192289122095454998343960170131050507959205808799577983896925995399341053579669906298422761291420625929575809148448578562225866063649832648992458579304776853597785371086357075950097253799298148915420003811660844595658233874696091881670559594966955891204928673588510886926483431284891512448657991374865775488000 = 2^10 3^5 5^3 7^2 11 13 17 19 29 31 37 41 43 61 71 73 97 101 109 113 127 151 163 181 193 211 241 257 271 281 337 379 401 421 433 449 541 577 601 631 641 673 701 757 769 811 1009 1051 1153 1201 1297 1601 1621 1801 2017 2161 2269 2521 2593 2689 2801 3361 3457 4051 4201 4481 4801 6301 6481 7561 8101 8641 9601 10369 12097 12601 13441 14401 15121 20161 21601 26881 28351 30241 32401 33601 43201 45361 56701 57601 72577 100801 103681 134401 151201 172801 241921 259201 453601 604801
  log n  743.361
Euler Phi 8539317693061962629180697769962974520054525126279991560831278388330352490426583586512992900188496112176901498474416638428268318831959091444137462535489639483874997182706914293258049826352519244684837433066725544581824417628160000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 = 2^406 3^195 5^97 7^46
  Euler Phi / Carmichael 2353207036227392699840359835197027810861586509667105258165585975620136819451770168241014357415260172006421268318567195334068650471770031813309485927989869787223048165428492695452504912464869721308652290858334861271446323200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 = 2^398 3^191 5^95 7^45
 log ( Euler Phi / Carmichael)   726.17
 log ( Euler Phi / Carmichael) / log n  0.976874
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39916800
 carm  39916800 = 2^8 3^4 5^2 7 11
 n 34719461278787479788276107325851819256329372628301543030133529928327224977259861239561378978102050970832090298448735600843363452131295995067565926756087143109966522956129218420264978659475620447899209060728637736185644345879767408170968369719573443918090563273423249907532394956247720224605046456766091349937058197033896481337235315216974562885972777146246393502148769034491990703126332866917536041480289977350155416582985529296871605957906929997675514815171359968542324519795972456175626868000917606741759680918649443963385779780352708508524808487070549137186877224498233069132443128345664796681868459855639903111582540465671549526451545498292706894188048988559803643008000 = 2^10 3^5 5^3 7^2 11^2 13 17 19 23 29 31 37 41 43 61 67 71 73 89 97 101 109 113 127 151 163 181 193 199 211 241 257 271 281 331 337 353 379 397 401 421 433 449 463 541 577 601 617 631 641 661 673 701 757 769 811 881 991 1009 1051 1153 1201 1297 1321 1409 1601 1621 1783 1801 2017 2113 2161 2269 2311 2377 2521 2593 2689 2801 2971 3169 3301 3361 3457 3697 3851 4051 4159 4201 4481 4621 4801 4951 5281 6301 6337 6481 7129 7393 7561 8101 8317 8641 9241 9601 9857 9901 10369 11551 12097 12601 13441 14081 14401 14851 15121 15401 16633 18481 19009 19801 20161 21121 21601 23761 26881 28351 28513 29569 30241 32401 33601 34651 35201 43201 45361 47521 55441 56701 57601 63361 66529 72577 79201 89101 92401 98561 99793 100801 103681 103951 105601 110881 118801 134401 151201 172801 177409 228097 241921 259201 285121 316801 332641 415801 453601 498961 604801 739201 950401 1108801 1247401 1425601 1995841 3991681 5702401 6652801 7983361 13305601 39916801
  log n  1550.88
Euler Phi 3903206617822737603606800357640106248123536511470428510022818920431166421482576035580553079061527205343171307787232985732854788251295341953805460019296695941170600869353460827271596963383747832512598610056136748944083965124400029587596192393177423091072131944066627433040051461258454889674955050662466427238284556892589679870882855079103239919161411862167994533532945642160833684481015028998521526495092989934660526983989245623282285738535082399230455941471744910606857018301819255633522769421926400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 = 2^745 3^358 5^176 7^83 11^81
  Euler Phi / Carmichael 97783555240468614808972672098968510705355552335618799854267349096900714022230640622007602790342091684282590482885225913220869114039585887491118025976448411224612215141330488097031750124853390865815862244872754052030322198282428190325782437299017533746997052470804960143098932310667560768271881780665444806153914063566961276226622752302369927428085714841069287456232604872154924354683116607506651998534276042534985945366092613217549646728572490761545412995824938637537503464752165895901544448000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 = 2^737 3^354 5^174 7^82 11^80
 log ( Euler Phi / Carmichael)   1531.2
 log ( Euler Phi / Carmichael) / log n  0.987305
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479001600
 carm  479001600 = 2^10 3^5 5^2 7 11
 n 38290447555320773414306597332873025464923833535197515188439665250832913116282699987647016765666329099848978482695072295384834028156546028943670044388361706310646543527065127845573047940443492332459630920719849973433681772514518152968482681887727853707545126493624592246011371133509748288650440181222912308223618021496118735395254675070266971741678883427477392517745610308093190197303680234332165374596615897958585604842241832577733736759685293481067065224264843443962501495170922844544749469495158371274830381387104410312001168005300098702505324482263996335814001290576076238256024099511487061931110991779366354006979377158823778900506013178643311997923338928039626971604436251824556059138265119244177208670987508841119016511925368685919684223053729750328324990936716364079119848646446029363789950800711743941474004647940201276018649407645405971751530081165809536428286152196814048526462612520909415387102330026404413522782966052071228688238252194863344484619568079083151813922728136860503488042404852112461480557565392173611250559157752311854592000 = 2^12 3^6 5^3 7^2 11^2 13 17 19 23 29 31 37 41 43 61 67 71 73 89 97 101 109 113 127 151 163 181 193 199 211 241 257 271 281 331 337 353 379 397 401 421 433 449 463 487 541 577 601 617 631 641 661 673 701 757 769 811 881 991 1009 1051 1153 1201 1297 1321 1409 1601 1621 1783 1801 2017 2113 2161 2269 2311 2377 2521 2593 2689 2801 2971 3169 3301 3361 3457 3697 3851 3889 4051 4159 4201 4481 4621 4801 4861 4951 5281 5347 6301 6337 6481 7129 7393 7561 7681 8101 8317 8641 9241 9601 9721 9857 9901 10369 10753 11551 12097 12601 13441 14081 14401 14851 15121 15361 15401 16633 17011 17921 18481 19009 19441 19801 20161 21121 21601 23041 23761 25601 26731 26881 28351 28513 29569 30241 32257 32401 33601 34651 35201 37423 43201 45361 47521 55441 56701 57601 63361 64513 66529 68041 72577 76801 77761 79201 84481 89101 92401 96769 98561 99793 100801 101377 103681 103951 105601 106921 110881 115201 118273 118801 134401 138241 149689 151201 155521 161281 170101 172801 177409 187111 228097 241921 259201 267301 285121 311041 316801 332641 340201 345601 394241 414721 415801 427681 453601 456193 498961 534601 604801 680401 684289 739201 748441 760321 950401 985601 1088641 1108801 1247401 1425601 1451521 1995841 2073601 2138401 2177281 2395009 2534401 2903041 3548161 3742201 3991681 4354561 5702401 5913601 6652801 6842881 7484401 7603201 7983361 8709121 8870401 10886401 13305601 22809601 29937601 39916801 47900161 59875201
  log n  2414.45
Euler Phi 4289066813038320475473541078133155574831062141203107568869074050479844460710489216717072564499789438242250368176358441343243899697990283839514481788266980965033409859933167903745014786157374525703917728327475046494563402309520842630455287199734090332428705286623066056367731779039258467281359643954799897388439771863289307571323633507859990463185519937644170566351597450515506217374756041021050533820272411967531163306045670445885276648180747481583373370512193826802708743681840057441521228090863423425913804905838281343233764274821503733200752789834505245917031573067551373387211397743531180221778465518921148559120505034149220910343209579268171869815812648176271521646672148623197969107854931344813244022328647297523678900959559902903418650801765607696844255362523701372220039875452994492301312000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 = 2^1212 3^608 5^252 7^117 11^113
  Euler Phi / Carmichael 8954180556053091420724985215358686849545099935372047961570637865259415544145341511838525308683289237952963764998610529366173097747461143844852463516336857674449124720947002898831684040632378943418806384628934530687503762637788355259054014015264438224065859668575357694771232035632570887615739997433828816831592570595357734862104079626999138339382415293903340962434358153533320593030912717245726389682774362272550161222938859590208626961122358425490381181424433293756657062694237466934392762134538639173467906799973698090431773661761262870939789741484173008852228412321694485753724826271000306098723815367049188476866267323844473401222896915726736340370914519233905526926574250739868027805867310975189318829683757418605029505036225146019175407350968363564639983170251835008943685940616888320000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 = 2^1202 3^603 5^250 7^116 11^112
 log ( Euler Phi / Carmichael)   2392.28
 log ( Euler Phi / Carmichael) / log n  0.990815
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2
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This is conditional on Dickson's conjecture, but may be of interest.

You may choose any admissible prime k-tuple $(b_1=0,b_2,b_3,\ldots,b_k)$. By definition the $b_i$ avoid some residue modulo every prime, and hence so do $\{(2+b_i)n+1\}$. Then it is a consequence of Dickson's conjecture that $q_i=(2+b_i)n+1$ are simultaneously prime for $1\le i \le k$ for infinitely many choices of $n$.

Write $N=\prod q_i$, then $$ \begin{align} N &= \prod_{i=1}^k\left((2+b_i)n+1\right) \\ & < (n+1)^k\prod_{i=1}^k (2+b_i) \\ &= C_1 (n+1)^k \end{align} $$ and $$ \begin{align} \varphi(N)&=\prod_{i=1}^k (q_i-1) \\ &= n^k \prod_{i=1}^k (2+b_i) \\ &= C_1 n^k \end{align} $$ and $$ \begin{align} \lambda(N) &= \operatorname{lcm} \left(2n,(2+b_2)n,\ldots,(2+b_k)n\right) \\ & \le n\operatorname{lcm}\left(2+b_2,2+b_3,\ldots,2+b_k\right) \\ &= C_2 n \end{align} $$ Where $C_1$ and $C_2$ depend on the choice of k-tuple but not on $n$. Thus $$ f(N) = \frac{\varphi(N)}{\lambda(N)} \ge \frac{C_1}{C_2}n^{k-1} = \Theta\left(N^{1-1/k}\right) $$ and by appropriate choice of $k$ can be made to grow faster than $N^{1-\epsilon}$ for any $\epsilon>0$.

For example, $(0,2,6,12,14,20,24,26,30,32)$ is an admissible 10-tuple of prime differences. We find simultaneous primes $q_i$ when $$ n = 190103265 \\ N \approx 1.540\times10^{94} \\ \varphi(N) \approx 1.540\times 10^{94} \\ \lambda(N) = 544544n \approx 1.035\times10^{14}\\ f(N) \approx 1.488\times10^{80} \\ \frac{\log f(N)}{\log N} = 0.8512\ldots $$ and when $$ n = 5092580565 \\ N \approx 2.93\times10^{108} \\ \varphi(N) \approx 2.93\times 10^{108} \\ \lambda(N) = 544544n \approx 2.77\times10^{15}\\ f(N) \approx 1.057\times10^{93} \\ \frac{\log f(N)}{\log N} = 0.8576\ldots $$

For larger $n$ that yield solutions we will have $\log_N f \to 9/10$. Similarly, for a larger choice of $k$ we should be able to make $\log_N f \to 1-1/k$, though in practice it is challenging to find examples.

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1
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One thing you definitely want to do is simply find the sequence of integers for which your quantity increases, and factor those. This is the entire story for a number of optimization problems that go back to Ramanujan. Those are all multiplicative functions. The bad news for you is that the Carmichael function is not multiplicative.

Please take a look at the table below. It seems correct, compared with the OEIS lists of totient and Carmichael values.

Notice the extent to which the two or three largest prime factors are $1 \pmod 3.$ This fits up very well with the largest prime factor being exactly one larger than the Carmichael value, as the overall Carmichael value is not the product over the prime power factors, it is the least common multiple.

8 = 2^3       phi 4 carm 2  ratio 2
24 = 2^3 * 3       phi 8 carm 2  ratio 4
63 = 3^2 * 7       phi 36 carm 6  ratio 6
80 = 2^4 * 5       phi 32 carm 4  ratio 8
240 = 2^4 * 3 * 5       phi 64 carm 4  ratio 16
455 = 5 * 7 * 13       phi 288 carm 12  ratio 24
819 = 3^2 * 7 * 13       phi 432 carm 12  ratio 36
1365 = 3 * 5 * 7 * 13       phi 576 carm 12  ratio 48
2387 = 7 * 11 * 31       phi 1800 carm 30  ratio 60
2720 = 2^5 * 5 * 17       phi 1024 carm 16  ratio 64
3276 = 2^2 * 3^2 * 7 * 13       phi 864 carm 12  ratio 72
3591 = 3^3 * 7 * 19       phi 1944 carm 18  ratio 108
4095 = 3^2 * 5 * 7 * 13       phi 1728 carm 12  ratio 144
7280 = 2^4 * 5 * 7 * 13       phi 2304 carm 12  ratio 192
9139 = 13 * 19 * 37       phi 7776 carm 36  ratio 216
13104 = 2^4 * 3^2 * 7 * 13       phi 3456 carm 12  ratio 288
18981 = 3^3 * 19 * 37       phi 11664 carm 36  ratio 324
21483 = 3^2 * 7 * 11 * 31       phi 10800 carm 30  ratio 360
21840 = 2^4 * 3 * 5 * 7 * 13       phi 4608 carm 12  ratio 384
24605 = 5 * 7 * 19 * 37       phi 15552 carm 36  ratio 432
32760 = 2^3 * 3^2 * 5 * 7 * 13       phi 6912 carm 12  ratio 576
44289 = 3^2 * 7 * 19 * 37       phi 23328 carm 36  ratio 648
45695 = 5 * 13 * 19 * 37       phi 31104 carm 36  ratio 864
63973 = 7 * 13 * 19 * 37       phi 46656 carm 36  ratio 1296
122915 = 5 * 13 * 31 * 61       phi 86400 carm 60  ratio 1440
132867 = 3^3 * 7 * 19 * 37       phi 69984 carm 36  ratio 1944
172081 = 7 * 13 * 31 * 61       phi 129600 carm 60  ratio 2160
191919 = 3 * 7 * 13 * 19 * 37       phi 93312 carm 36  ratio 2592
246753 = 3^3 * 13 * 19 * 37       phi 139968 carm 36  ratio 3888
319865 = 5 * 7 * 13 * 19 * 37       phi 186624 carm 36  ratio 5184
520025 = 5^2 * 11 * 31 * 61       phi 360000 carm 60  ratio 6000
575757 = 3^2 * 7 * 13 * 19 * 37       phi 279936 carm 36  ratio 7776
860405 = 5 * 7 * 13 * 31 * 61       phi 518400 carm 60  ratio 8640
959595 = 3 * 5 * 7 * 13 * 19 * 37       phi 373248 carm 36  ratio 10368
1233765 = 3^3 * 5 * 13 * 19 * 37       phi 559872 carm 36  ratio 15552
1727271 = 3^3 * 7 * 13 * 19 * 37       phi 839808 carm 36  ratio 23328
2878785 = 3^2 * 5 * 7 * 13 * 19 * 37       phi 1119744 carm 36  ratio 31104
3640175 = 5^2 * 7 * 11 * 31 * 61       phi 2160000 carm 60  ratio 36000
4302025 = 5^2 * 7 * 13 * 31 * 61       phi 2592000 carm 60  ratio 43200
4670029 = 7 * 13 * 19 * 37 * 73       phi 3359232 carm 72  ratio 46656
6760325 = 5^2 * 11 * 13 * 31 * 61       phi 4320000 carm 60  ratio 72000
8636355 = 3^3 * 5 * 7 * 13 * 19 * 37       phi 3359232 carm 36  ratio 93312 

Alrighty, after seeing how well 36, 60, and 72 do, I decided to turn it around. For each Carmichael number, examine each prime power that can divide $n$ and still divide the carmichael number. Then I found the ratio $f(n)$ and sorted by $\log f(n) / \log n$

0.662224carm  80 = 2^4 * 5 ratio  102400  n factored  = 2^6 3 5^2 11 17 41
0.699123carm  96 = 2^5 * 3 ratio  1769472  n factored  = 2^7 3^2 5 7 13 17 97
0.700197carm  84 = 2^2 * 3 * 7 ratio  1354752  n factored  = 2^4 3^2 5 7^2 13 29 43
0.721462carm  36 = 2^2 * 3^2 ratio  746496  n factored  = 2^4 3^3 5 7 13 19 37
0.744067carm  60 = 2^2 * 3 * 5 ratio  20736000  n factored  = 2^4 3^2 5^2 7 11 13 31 61
0.750171carm  72 = 2^3 * 3^2 ratio  53747712  n factored  = 2^5 3^3 5 7 13 19 37 73
....
0.800189carm  10860 = 2^2 * 3 * 5 * 181
0.800339carm  11124 = 2^2 * 3^3 * 103
0.800662carm  10008 = 2^3 * 3^2 * 139
0.800797carm  432 = 2^4 * 3^3
0.800881carm  4050 = 2 * 3^4 * 5^2
0.801043carm  10626 = 2 * 3 * 7 * 11 * 23
0.801421carm  9540 = 2^2 * 3^2 * 5 * 53
0.801459carm  6400 = 2^8 * 5^2
0.801485carm  3480 = 2^3 * 3 * 5 * 29
0.801868carm  3640 = 2^3 * 5 * 7 * 13
0.802727carm  10640 = 2^4 * 5 * 7 * 19
0.802903carm  10908 = 2^2 * 3^3 * 101
0.802914carm  10728 = 2^3 * 3^2 * 149
0.803203carm  1764 = 2^2 * 3^2 * 7^2
0.803336carm  180 = 2^2 * 3^2 * 5
0.803402carm  5928 = 2^3 * 3 * 13 * 19
0.803579carm  6440 = 2^3 * 5 * 7 * 23
0.803747carm  8220 = 2^2 * 3 * 5 * 137
0.804008carm  10164 = 2^2 * 3 * 7 * 11^2
0.804178carm  9750 = 2 * 3 * 5^3 * 13
0.804402carm  7968 = 2^5 * 3 * 83
0.804826carm  8000 = 2^6 * 5^3
0.805426carm  5200 = 2^4 * 5^2 * 13
0.805766carm  8256 = 2^6 * 3 * 43
0.805773carm  5250 = 2 * 3 * 5^3 * 7
0.806533carm  8748 = 2^2 * 3^7
0.806591carm  11376 = 2^4 * 3^2 * 79
0.806628carm  4104 = 2^3 * 3^3 * 19
0.806669carm  780 = 2^2 * 3 * 5 * 13
0.807051carm  480 = 2^5 * 3 * 5
0.807705carm  7296 = 2^7 * 3 * 19
0.808179carm  7668 = 2^2 * 3^3 * 71
0.808211carm  6528 = 2^7 * 3 * 17
0.808296carm  8040 = 2^3 * 3 * 5 * 67
0.808322carm  990 = 2 * 3^2 * 5 * 11
0.808589carm  864 = 2^5 * 3^3
0.808876carm  4760 = 2^3 * 5 * 7 * 17
0.809143carm  11440 = 2^4 * 5 * 11 * 13
0.809343carm  1836 = 2^2 * 3^3 * 17
0.810122carm  3036 = 2^2 * 3 * 11 * 23
0.810335carm  1500 = 2^2 * 3 * 5^3
0.810376carm  6372 = 2^2 * 3^3 * 59
0.810402carm  6156 = 2^2 * 3^4 * 19
0.810433carm  9840 = 2^4 * 3 * 5 * 41
0.810533carm  9648 = 2^4 * 3^2 * 67
0.810578carm  9282 = 2 * 3 * 7 * 13 * 17
0.810626carm  11328 = 2^6 * 3 * 59
0.810734carm  5256 = 2^3 * 3^2 * 73
0.810739carm  5508 = 2^2 * 3^4 * 17
0.810954carm  2040 = 2^3 * 3 * 5 * 17
0.810961carm  6360 = 2^3 * 3 * 5 * 53
0.810991carm  11136 = 2^7 * 3 * 29
0.811092carm  6270 = 2 * 3 * 5 * 11 * 19
0.81113carm  5850 = 2 * 3^2 * 5^2 * 13
0.811364carm  5076 = 2^2 * 3^3 * 47
0.81164carm  4704 = 2^5 * 3 * 7^2
0.811998carm  1932 = 2^2 * 3 * 7 * 23
0.812192carm  5640 = 2^3 * 3 * 5 * 47
0.813107carm  5328 = 2^4 * 3^2 * 37
0.813474carm  7040 = 2^7 * 5 * 11
0.813487carm  6132 = 2^2 * 3 * 7 * 73
0.813667carm  7840 = 2^5 * 5 * 7^2
0.813889carm  10980 = 2^2 * 3^2 * 5 * 61
0.813914carm  5346 = 2 * 3^5 * 11
0.814096carm  11480 = 2^3 * 5 * 7 * 41
0.814271carm  630 = 2 * 3^2 * 5 * 7
0.814697carm  7000 = 2^3 * 5^3 * 7
0.814905carm  11466 = 2 * 3^2 * 7^2 * 13
0.814916carm  9048 = 2^3 * 3 * 13 * 29
0.815185carm  660 = 2^2 * 3 * 5 * 11
0.815302carm  10098 = 2 * 3^3 * 11 * 17
0.815409carm  8700 = 2^2 * 3 * 5^2 * 29
0.815849carm  900 = 2^2 * 3^2 * 5^2
0.815864carm  11070 = 2 * 3^3 * 5 * 41
0.816586carm  960 = 2^6 * 3 * 5
0.816601carm  11100 = 2^2 * 3 * 5^2 * 37
0.817089carm  7280 = 2^4 * 5 * 7 * 13
0.817844carm  600 = 2^3 * 3 * 5^2
0.818152carm  1584 = 2^4 * 3^2 * 11
0.819568carm  2112 = 2^6 * 3 * 11
0.819669carm  576 = 2^6 * 3^2
0.820554carm  792 = 2^3 * 3^2 * 11
0.820789carm  5568 = 2^6 * 3 * 29
0.820828carm  10400 = 2^5 * 5^2 * 13
0.820878carm  3420 = 2^2 * 3^2 * 5 * 19
0.821008carm  4788 = 2^2 * 3^2 * 7 * 19
0.821349carm  9996 = 2^2 * 3 * 7^2 * 17
0.821948carm  1248 = 2^5 * 3 * 13
0.822682carm  4590 = 2 * 3^3 * 5 * 17
0.822732carm  7992 = 2^3 * 3^3 * 37
0.82276carm  4158 = 2 * 3^3 * 7 * 11
0.822772carm  9744 = 2^4 * 3 * 7 * 29
0.822828carm  2916 = 2^2 * 3^6
0.823318carm  10764 = 2^2 * 3^2 * 13 * 23
0.823517carm  7452 = 2^2 * 3^4 * 23
0.82354carm  6160 = 2^4 * 5 * 7 * 11
0.824167carm  2088 = 2^3 * 3^2 * 29
0.824257carm  672 = 2^5 * 3 * 7
0.824676carm  5220 = 2^2 * 3^2 * 5 * 29
0.824967carm  1092 = 2^2 * 3 * 7 * 13
0.825046carm  8928 = 2^5 * 3^2 * 31
0.82518carm  7812 = 2^2 * 3^2 * 7 * 31
0.825341carm  4950 = 2 * 3^2 * 5^2 * 11
0.825371carm  3150 = 2 * 3^2 * 5^2 * 7
0.825485carm  360 = 2^3 * 3^2 * 5
0.825763carm  3864 = 2^3 * 3 * 7 * 23
0.825794carm  5832 = 2^3 * 3^6
0.825813carm  3564 = 2^2 * 3^4 * 11
0.825949carm  1656 = 2^3 * 3^2 * 23
0.826019carm  8760 = 2^3 * 3 * 5 * 73
0.827899carm  10620 = 2^2 * 3^2 * 5 * 59
0.827935carm  1920 = 2^7 * 3 * 5
0.828085carm  5160 = 2^3 * 3 * 5 * 43
0.828107carm  8112 = 2^4 * 3 * 13^2
0.828157carm  9324 = 2^2 * 3^2 * 7 * 37
0.828487carm  2496 = 2^6 * 3 * 13
0.828603carm  9856 = 2^7 * 7 * 11
0.828624carm  10224 = 2^4 * 3^2 * 71
0.82872carm  5148 = 2^2 * 3^2 * 11 * 13
0.828728carm  10416 = 2^4 * 3 * 7 * 31
0.828831carm  4830 = 2 * 3 * 5 * 7 * 23
0.829213carm  1224 = 2^3 * 3^2 * 17
0.829347carm  9396 = 2^2 * 3^4 * 29
0.829402carm  4480 = 2^7 * 5 * 7
0.829983carm  6006 = 2 * 3 * 7 * 11 * 13
0.83006carm  6384 = 2^4 * 3 * 7 * 19
0.830184carm  7416 = 2^3 * 3^2 * 103
0.830446carm  1296 = 2^4 * 3^4
0.830631carm  4056 = 2^3 * 3 * 13^2
0.831055carm  2772 = 2^2 * 3^2 * 7 * 11
0.8311carm  5112 = 2^3 * 3^2 * 71
0.831637carm  7752 = 2^3 * 3 * 17 * 19
0.831928carm  4224 = 2^7 * 3 * 11
0.832678carm  2184 = 2^3 * 3 * 7 * 13
0.832744carm  8424 = 2^3 * 3^4 * 13
0.832778carm  9300 = 2^2 * 3 * 5^2 * 31
0.833027carm  1152 = 2^7 * 3^2
0.833463carm  4440 = 2^3 * 3 * 5 * 37
0.833886carm  10944 = 2^6 * 3^2 * 19
0.833933carm  2808 = 2^3 * 3^3 * 13
0.833969carm  5100 = 2^2 * 3 * 5^2 * 17
0.834267carm  6864 = 2^4 * 3 * 11 * 13
0.83432carm  7590 = 2 * 3 * 5 * 11 * 23
0.834646carm  3528 = 2^3 * 3^2 * 7^2
0.834709carm  8520 = 2^3 * 3 * 5 * 71
0.834914carm  420 = 2^2 * 3 * 5 * 7
0.834963carm  2448 = 2^4 * 3^2 * 17
0.834974carm  10152 = 2^3 * 3^3 * 47
0.835087carm  8960 = 2^8 * 5 * 7
0.835185carm  7956 = 2^2 * 3^2 * 13 * 17
0.835672carm  10512 = 2^4 * 3^2 * 73
0.836432carm  756 = 2^2 * 3^3 * 7
0.836452carm  5304 = 2^3 * 3 * 13 * 17
0.836717carm  1890 = 2 * 3^3 * 5 * 7
0.836732carm  936 = 2^3 * 3^2 * 13
0.837079carm  3432 = 2^3 * 3 * 11 * 13
0.837438carm  2310 = 2 * 3 * 5 * 7 * 11
0.837487carm  5600 = 2^5 * 5^2 * 7
0.837495carm  9216 = 2^10 * 3^2
0.837593carm  4896 = 2^5 * 3^2 * 17
0.837616carm  2970 = 2 * 3^3 * 5 * 11
0.837663carm  2730 = 2 * 3 * 5 * 7 * 13
0.837676carm  7440 = 2^4 * 3 * 5 * 31
0.837709carm  7350 = 2 * 3 * 5^2 * 7^2
0.837813carm  1728 = 2^6 * 3^3
0.838364carm  1512 = 2^3 * 3^3 * 7
0.839819carm  9576 = 2^3 * 3^2 * 7 * 19
0.839956carm  6072 = 2^3 * 3 * 11 * 23
0.840315carm  11484 = 2^2 * 3^2 * 11 * 29
0.84062carm  10608 = 2^4 * 3 * 13 * 17
0.840797carm  720 = 2^4 * 3^2 * 5
0.840984carm  10332 = 2^2 * 3^2 * 7 * 41
0.84111carm  3276 = 2^2 * 3^2 * 7 * 13
0.841276carm  8892 = 2^2 * 3^2 * 13 * 19
0.841741carm  9450 = 2 * 3^3 * 5^2 * 7
0.841893carm  10140 = 2^2 * 3 * 5 * 13^2
0.84201carm  10656 = 2^5 * 3^2 * 37
0.842161carm  2760 = 2^3 * 3 * 5 * 23
0.842208carm  4992 = 2^7 * 3 * 13
0.842353carm  6264 = 2^3 * 3^3 * 29
0.842503carm  2376 = 2^3 * 3^3 * 11
0.842553carm  1380 = 2^2 * 3 * 5 * 23
0.842623carm  6960 = 2^4 * 3 * 5 * 29
0.843108carm  1440 = 2^5 * 3^2 * 5
0.843183carm  11220 = 2^2 * 3 * 5 * 11 * 17
0.843403carm  7320 = 2^3 * 3 * 5 * 61
0.843785carm  7380 = 2^2 * 3^2 * 5 * 41
0.844698carm  4176 = 2^4 * 3^2 * 29
0.84545carm  10530 = 2 * 3^4 * 5 * 13
0.845624carm  3300 = 2^2 * 3 * 5^2 * 11
0.845831carm  3312 = 2^4 * 3^2 * 23
0.845858carm  7308 = 2^2 * 3^2 * 7 * 29
0.845866carm  7260 = 2^2 * 3 * 5 * 11^2
0.846269carm  4608 = 2^9 * 3^2
0.846425carm  11280 = 2^4 * 3 * 5 * 47
0.846695carm  4968 = 2^3 * 3^3 * 23
0.846796carm  7740 = 2^2 * 3^2 * 5 * 43
0.846831carm  2800 = 2^4 * 5^2 * 7
0.847086carm  8208 = 2^4 * 3^3 * 19
0.847402carm  6624 = 2^5 * 3^2 * 23
0.847522carm  5796 = 2^2 * 3^2 * 7 * 23
0.847569carm  1560 = 2^3 * 3 * 5 * 13
0.8478carm  4410 = 2 * 3^2 * 5 * 7^2
0.847802carm  540 = 2^2 * 3^3 * 5
0.848407carm  1344 = 2^6 * 3 * 7
0.848697carm  10692 = 2^2 * 3^5 * 11
0.84873carm  6426 = 2 * 3^3 * 7 * 17
0.849121carm  2700 = 2^2 * 3^3 * 5^2
0.84946carm  6732 = 2^2 * 3^2 * 11 * 17
0.849536carm  7728 = 2^4 * 3 * 7 * 23
0.849864carm  7980 = 2^2 * 3 * 5 * 7 * 19
0.850001carm  2856 = 2^3 * 3 * 7 * 17
0.850121carm  4356 = 2^2 * 3^2 * 11^2
0.850147carm  4500 = 2^2 * 3^2 * 5^3
0.850498carm  3840 = 2^8 * 3 * 5
0.851181carm  8460 = 2^2 * 3^2 * 5 * 47
0.851231carm  840 = 2^3 * 3 * 5 * 7
0.851354carm  5670 = 2 * 3^4 * 5 * 7
0.851511carm  1008 = 2^4 * 3^2 * 7
0.851869carm  7176 = 2^3 * 3 * 13 * 23
0.852089carm  1320 = 2^3 * 3 * 5 * 11
0.852111carm  8448 = 2^8 * 3 * 11
0.852278carm  3696 = 2^4 * 3 * 7 * 11
0.852417carm  4080 = 2^4 * 3 * 5 * 17
0.853027carm  3888 = 2^4 * 3^5
0.853195carm  3000 = 2^3 * 3 * 5^3
0.853296carm  4752 = 2^4 * 3^3 * 11
0.853308carm  2280 = 2^3 * 3 * 5 * 19
0.853319carm  10836 = 2^2 * 3^2 * 7 * 43
0.853336carm  1872 = 2^4 * 3^2 * 13
0.853336carm  9408 = 2^6 * 3 * 7^2
0.853647carm  2592 = 2^5 * 3^4
0.853916carm  6804 = 2^2 * 3^5 * 7
0.854251carm  7644 = 2^2 * 3 * 7^2 * 13
0.854473carm  2268 = 2^2 * 3^4 * 7
0.854735carm  4536 = 2^3 * 3^4 * 7
0.854835carm  9792 = 2^6 * 3^2 * 17
0.855229carm  2304 = 2^8 * 3^2
0.855401carm  4368 = 2^4 * 3 * 7 * 13
0.855572carm  3900 = 2^2 * 3 * 5^2 * 13
0.855604carm  4284 = 2^2 * 3^2 * 7 * 17
0.856454carm  2940 = 2^2 * 3 * 5 * 7^2
0.856974carm  6660 = 2^2 * 3^2 * 5 * 37
0.85737carm  1080 = 2^3 * 3^3 * 5
0.857442carm  2688 = 2^7 * 3 * 7
0.857869carm  5544 = 2^3 * 3^2 * 7 * 11
0.858177carm  3168 = 2^5 * 3^2 * 11
0.858373carm  11200 = 2^6 * 5^2 * 7
0.858567carm  10320 = 2^4 * 3 * 5 * 43
0.858768carm  5292 = 2^2 * 3^3 * 7^2
0.859203carm  6900 = 2^2 * 3 * 5^2 * 23
0.8598carm  11016 = 2^3 * 3^4 * 17
0.860214carm  7680 = 2^9 * 3 * 5
0.860511carm  2880 = 2^6 * 3^2 * 5
0.861515carm  10890 = 2 * 3^2 * 5 * 11^2
0.861706carm  5616 = 2^4 * 3^3 * 13
0.861761carm  5700 = 2^2 * 3 * 5^2 * 19
0.862782carm  10440 = 2^3 * 3^2 * 5 * 29
0.863074carm  3672 = 2^3 * 3^3 * 17
0.863215carm  10260 = 2^2 * 3^3 * 5 * 19
0.863755carm  3456 = 2^7 * 3^3
0.865186carm  9520 = 2^4 * 5 * 7 * 17
0.865187carm  2340 = 2^2 * 3^2 * 5 * 13
0.865273carm  1800 = 2^3 * 3^2 * 5^2
0.865527carm  8910 = 2 * 3^4 * 5 * 11
0.866693carm  2400 = 2^5 * 3 * 5^2
0.866712carm  1200 = 2^4 * 3 * 5^2
0.867314carm  4140 = 2^2 * 3^2 * 5 * 23
0.867705carm  11040 = 2^5 * 3 * 5 * 23
0.867788carm  5520 = 2^4 * 3 * 5 * 23
0.867791carm  5184 = 2^6 * 3^4
0.868045carm  3744 = 2^5 * 3^2 * 13
0.868129carm  7776 = 2^5 * 3^5
0.868161carm  3060 = 2^2 * 3^2 * 5 * 17
0.868433carm  10200 = 2^3 * 3 * 5^2 * 17
0.86899carm  2100 = 2^2 * 3 * 5^2 * 7
0.869346carm  8160 = 2^5 * 3 * 5 * 17
0.869884carm  8880 = 2^4 * 3 * 5 * 37
0.869923carm  9936 = 2^4 * 3^3 * 23
0.871505carm  3120 = 2^4 * 3 * 5 * 13
0.871633carm  5376 = 2^8 * 3 * 7
0.871888carm  5712 = 2^4 * 3 * 7 * 17
0.871994carm  7128 = 2^3 * 3^4 * 11
0.872133carm  10584 = 2^3 * 3^3 * 7^2
0.872429carm  5580 = 2^2 * 3^2 * 5 * 31
0.872687carm  2640 = 2^4 * 3 * 5 * 11
0.872761carm  9984 = 2^8 * 3 * 13
0.873239carm  11232 = 2^5 * 3^3 * 13
0.873295carm  9108 = 2^2 * 3^2 * 11 * 23
0.873935carm  1260 = 2^2 * 3^2 * 5 * 7
0.874132carm  8712 = 2^3 * 3^2 * 11^2
0.874243carm  7344 = 2^4 * 3^3 * 17
0.874661carm  8352 = 2^5 * 3^2 * 29
0.874757carm  1620 = 2^2 * 3^4 * 5
0.875159carm  1980 = 2^2 * 3^2 * 5 * 11
0.875252carm  4560 = 2^4 * 3 * 5 * 19
0.875642carm  5760 = 2^7 * 3^2 * 5
0.876127carm  6912 = 2^8 * 3^3
0.876649carm  6930 = 2 * 3^2 * 5 * 7 * 11
0.876947carm  9504 = 2^5 * 3^3 * 11
0.877286carm  6600 = 2^3 * 3 * 5^2 * 11
0.877606carm  9828 = 2^2 * 3^3 * 7 * 13
0.878014carm  10296 = 2^3 * 3^2 * 11 * 13
0.878143carm  10752 = 2^9 * 3 * 7
0.87824carm  7140 = 2^2 * 3 * 5 * 7 * 17
0.878262carm  8580 = 2^2 * 3 * 5 * 11 * 13
0.87903carm  1680 = 2^4 * 3 * 5 * 7
0.879102carm  5880 = 2^3 * 3 * 5 * 7^2
0.87911carm  3240 = 2^3 * 3^4 * 5
0.879368carm  2016 = 2^5 * 3^2 * 7
0.879861carm  5400 = 2^3 * 3^3 * 5^2
0.880494carm  6552 = 2^3 * 3^2 * 7 * 13
0.880565carm  11424 = 2^5 * 3 * 7 * 17
0.881138carm  7056 = 2^4 * 3^2 * 7^2
0.881231carm  6000 = 2^4 * 3 * 5^3
0.881504carm  10710 = 2 * 3^2 * 5 * 7 * 17
0.881766carm  3024 = 2^4 * 3^3 * 7
0.881856carm  8190 = 2 * 3^2 * 5 * 7 * 13
0.882484carm  8280 = 2^3 * 3^2 * 5 * 23
0.883417carm  7800 = 2^3 * 3 * 5^2 * 13
0.883709carm  10500 = 2^2 * 3 * 5^3 * 7
0.883902carm  8316 = 2^2 * 3^3 * 7 * 11
0.884562carm  4860 = 2^2 * 3^5 * 5
0.88526carm  9120 = 2^5 * 3 * 5 * 19
0.885549carm  4320 = 2^5 * 3^3 * 5
0.88585carm  6840 = 2^3 * 3^2 * 5 * 19
0.886443carm  7392 = 2^5 * 3 * 7 * 11
0.886738carm  2160 = 2^4 * 3^3 * 5
0.887871carm  4620 = 2^2 * 3 * 5 * 7 * 11
0.888141carm  11160 = 2^3 * 3^2 * 5 * 31
0.888252carm  7200 = 2^5 * 3^2 * 5^2
0.889508carm  3600 = 2^4 * 3^2 * 5^2
0.889516carm  8736 = 2^5 * 3 * 7 * 13
0.889631carm  7020 = 2^2 * 3^3 * 5 * 13
0.88972carm  7488 = 2^6 * 3^2 * 13
0.89022carm  4800 = 2^6 * 3 * 5^2
0.890788carm  6240 = 2^5 * 3 * 5 * 13
0.890878carm  11088 = 2^4 * 3^2 * 7 * 11
0.891411carm  5280 = 2^5 * 3 * 5 * 11
0.891563carm  5460 = 2^2 * 3 * 5 * 7 * 13
0.892498carm  6336 = 2^6 * 3^2 * 11
0.892585carm  8568 = 2^3 * 3^2 * 7 * 17
0.893303carm  10368 = 2^7 * 3^4
0.893864carm  8100 = 2^2 * 3^4 * 5^2
0.89389carm  3960 = 2^3 * 3^2 * 5 * 11
0.895034carm  4200 = 2^3 * 3 * 5^2 * 7
0.895086carm  9072 = 2^4 * 3^4 * 7
0.895126carm  4032 = 2^6 * 3^2 * 7
0.895331carm  4680 = 2^3 * 3^2 * 5 * 13
0.895634carm  11400 = 2^3 * 3 * 5^2 * 19
0.895798carm  5940 = 2^2 * 3^3 * 5 * 11
0.895913carm  2520 = 2^3 * 3^2 * 5 * 7
0.896598carm  3360 = 2^5 * 3 * 5 * 7
0.896999carm  9720 = 2^3 * 3^5 * 5
0.897255carm  6048 = 2^5 * 3^3 * 7
0.897274carm  6120 = 2^3 * 3^2 * 5 * 17
0.898476carm  9000 = 2^3 * 3^2 * 5^3
0.898504carm  10560 = 2^6 * 3 * 5 * 11
0.898517carm  8820 = 2^2 * 3^2 * 5 * 7^2
0.898642carm  6300 = 2^2 * 3^2 * 5^2 * 7
0.900179carm  9600 = 2^7 * 3 * 5^2
0.90043carm  3780 = 2^2 * 3^3 * 5 * 7
0.901244carm  8640 = 2^6 * 3^3 * 5
0.901617carm  6720 = 2^6 * 3 * 5 * 7
0.901756carm  9900 = 2^2 * 3^2 * 5^2 * 11
0.902084carm  7920 = 2^4 * 3^2 * 5 * 11
0.903565carm  8064 = 2^7 * 3^2 * 7
0.905632carm  10920 = 2^3 * 3 * 5 * 7 * 13
0.906154carm  9180 = 2^2 * 3^3 * 5 * 17
0.908693carm  10800 = 2^4 * 3^3 * 5^2
0.909774carm  9660 = 2^2 * 3 * 5 * 7 * 23
0.910055carm  9360 = 2^4 * 3^2 * 5 * 13
0.911522carm  6480 = 2^4 * 3^4 * 5
0.912432carm  5040 = 2^4 * 3^2 * 5 * 7
0.913389carm  11340 = 2^2 * 3^4 * 5 * 7
0.914572carm  9240 = 2^3 * 3 * 5 * 7 * 11
0.918198carm  7560 = 2^3 * 3^3 * 5 * 7
0.920389carm  8400 = 2^4 * 3 * 5^2 * 7
0.923965carm  10080 = 2^5 * 3^2 * 5 * 7

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$\endgroup$

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