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I'm interested in a variation of https://oeis.org/A006339 (Least hypotenuse of n distinct Pythagorean triangles).

Basically, if A006339 is $u(n)$, I want $v(n) := \min_{n \leq i} u(i)$

In other words, $v(n)$ is increasing and for any $k \leq n$, there are less than $v(n)$ triples with hypothenuse $k$.

Are there tables for that sequence? How can I compute it?

A possible way would be to have lower bounds for A006339 and compute it until the lower bound is big enough. I'm not even sure about how I can compute A006339. There is an easy enumeration algorithm for Pythagorean triples due to a result by Hall (1970) and Roberts (1977) (https://stackoverflow.com/a/8263898/5133167), and it allows us to stop the search as any multiplication by $U$, $A$ or $D$ will increase the value of $c$. Is there a more performant way?

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  • $\begingroup$ Your title and first line disagree about "hipothenuse" or "hypotenuse" ( I think the last one is correct). $\endgroup$ Commented Oct 31, 2020 at 20:01
  • $\begingroup$ In the series A006339, the number $(1)$ is never a hypotenuse. I don't understand what you mean by "least hypotenuse" or "more than n distinct Pythagorean triples". I can help with finding triples by a dozen or more criteria but I don't understand these. $\endgroup$
    – poetasis
    Commented Nov 1, 2020 at 16:08
  • $\begingroup$ I do have a formula $F(n,k)$ where the hypotenuse is smaller for $n-k<1$ than for $n-k>1$ if that will help. $\endgroup$
    – poetasis
    Commented Nov 1, 2020 at 16:33
  • 2
    $\begingroup$ @poetasis I think the OP has walked away. I did answer with a method of Ramanujan suitable for programming this. Next, as I am interested, I will look at some of J.-L. Nicolas papers from 1971 to 1975. He puts his articles as pdf's on his department web page. I am looking at his 1988 survey in English, I think the entire "benefit" method will be revealed. G. Robin, his student, is more shy. math.univ-lyon1.fr/~nicolas/publications.html $\endgroup$
    – Will Jagy
    Commented Nov 1, 2020 at 16:54
  • $\begingroup$ Thanks for your comments! I fixed "hipothenuse". (1) is an edge case, I guess it's for the triplet (0,1,1)? I don't understand what your formula $F(n, k)$ does. @WillJagy I'll look at your answer now :) $\endgroup$
    – Labo
    Commented Nov 1, 2020 at 21:20

3 Answers 3

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Worked it out. Taking the consecutive (1 mod 4) primes $$ 5, 13, 17, 29, 37, 41, 53,...$$ we calculate the best exponent for each such prime $p$ in terms of the real positive exponent $\delta.$ As I said, this is Ramanujan's approach, used in his Superior Highly Composite Numbers. I learned it from Nicolas

Given real $\delta > 0,$ we demand the exponent of $p$ to be $$ k=\left\lfloor \frac{1}{p^\delta - 1} \right\rfloor $$ Meanwhile, with prime $p$ and desired exponent $k,$ we use $$ \delta = \frac{\log (k+1) - \log k}{\log p} $$ This defines a sequence of numbers that rapidly increase $r(n) .$ Indeed, $n$ is guaranteed to have more representations as the sum of two squares than any smaller number has.

As was Ramanujan's case, there may be new champions of $r(n)$ that lie between two of the numbers defined above. In that case, there is a procedure due to Guy Robin, I think in his dissertation (supervised by J.-L. Nicolas). I don't know how complicated his "benefit method" is, but I programmed such things (operations research) as an undergraduate, and recall that a fair amount of work was involved. The constraints include: the prime exponents are non-increasing and always non-negative.
Alright, Nicolas gives complete details of the "benefit" method in ACTA 1988 . The approach of Robin is to combine numerous tools. This is the article where he displays the smallest number with more than $10^{1000}$ divisors.

I can write a program to show the first few such numbers...

delta      r(n)           n = factored
0.3 ;;;; 4 times 2 ;;;; 5  =   5
0.27 ;;;; 4 times 4 ;;;; 65  =   5 13
0.25 ;;;; 4 times 6 ;;;; 325  =   5^2 13
0.24 ;;;; 4 times 12 ;;;; 5525  =   5^2 13 17
0.2 ;;;; 4 times 24 ;;;; 160225  =   5^2 13 17 29
0.19 ;;;; 4 times 48 ;;;; 5928325  =   5^2 13 17 29 37
0.18 ;;;; 4 times 96 ;;;; 243061325  =   5^2 13 17 29 37 41
0.175 ;;;; 4 times 128 ;;;; 1215306625  =   5^3 13 17 29 37 41
0.17 ;;;; 4 times 256 ;;;; 64411251125  =   5^3 13 17 29 37 41 53
0.168 ;;;; 4 times 512 ;;;; 3929086318625  =   5^3 13 17 29 37 41 53 61
0.161 ;;;; 4 times 1024 ;;;; 286823301259625  =   5^3 13 17 29 37 41 53 61 73
0.158 ;;;; 4 times 1536 ;;;; 3728702916375125  =   5^3 13^2 17 29 37 41 53 61 73
0.154 ;;;; 4 times 3072 ;;;; 331854559557386125  =   5^3 13^2 17 29 37 41 53 61 73 89
0.151 ;;;; 4 times 6144 ;;;; 32189892277066454125  =   5^3 13^2 17 29 37 41 53 61 73 89 97
0.15 ;;;; 4 times 12288 ;;;; 3251179119983711866625  =   5^3 13^2 17 29 37 41 53 61 73 89 97 101
0.147 ;;;; 4 times 24576 ;;;; 354378524078224593462125  =   5^3 13^2 17 29 37 41 53 61 73 89 97 101 109
0.146 ;;;; 4 times 49152 ;;;; 40044773220839379061220125  =   5^3 13^2 17 29 37 41 53 61 73 89 97 101 109 113
delta         r(n)                 n           =        factored

$$ \bigcirc \bigcirc \bigcirc \bigcirc\bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc\bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc\bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$

We get an ordered list of consecutive useful $\delta's$ by solving for each prime and each exponent up to $5.$ To make sure it works properly, let the $\delta$ used be in between two consecutive reals indicated in the list below. An alternative is to read the sorted file, ignore the actual value of $\delta$ on that line, and multiply the number $n$ by the prime on that line. If done properly, factoring the resulting $n$ at each step should show the increased exponent for that prime $p \; . \; \; $ $$ \bigcirc \bigcirc \bigcirc \bigcirc\bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc\bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc\bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$

 0.4306765580733931  5 ^ 1
  0.2702381544273197  13 ^ 1
  0.2519296364125923  5 ^ 2
   0.244650542118226  17 ^ 1
  0.2058468324604344  29 ^ 1
  0.1919587200065601  37 ^ 1
  0.1866524112389434  41 ^ 1
  0.1787469216608008  5 ^ 3
  0.1745834300480449  53 ^ 1
  0.1686130986895011  61 ^ 1
  0.1615554674429964  73 ^ 1
   0.158079186604075  13 ^ 2
  0.1544226628011101  89 ^ 1
  0.1515171524096389  97 ^ 1
   0.150190483223688  101 ^ 1
  0.1477501131786861  109 ^ 1
  0.1466237184553111  113 ^ 1
  0.1431113929202641  17 ^ 2
  0.1408841194731412  137 ^ 1
  0.1386468838532138  5 ^ 4
  0.1385201075671774  149 ^ 1
  0.1370873187823978  157 ^ 1
  0.1345057169479844  173 ^ 1
  0.1333360731748201  181 ^ 1
  0.1317096685686114  193 ^ 1
  0.1311982683517524  197 ^ 1
  0.1275639755045533  229 ^ 1
  0.1271587398372755  233 ^ 1
  0.1263760881150453  241 ^ 1
  0.1249121781636255  257 ^ 1
  0.1238932866290727  269 ^ 1
  0.1232476925609442  277 ^ 1
  0.1229342993142804  281 ^ 1
  0.1220292459879827  293 ^ 1
  0.1206269875212659  313 ^ 1
  0.1204126778815846  29 ^ 2
  0.1203610007213705  317 ^ 1
  0.1190957566850307  337 ^ 1
  0.1183840591148901  349 ^ 1
  0.1181540875758708  353 ^ 1
  0.1170544627527939  373 ^ 1
  0.1162300592682939  389 ^ 1
  0.1158346526511383  397 ^ 1
   0.115640914438019  401 ^ 1
  0.1152610595238196  409 ^ 1
  0.1147094651682719  421 ^ 1
  0.1141784098763921  433 ^ 1
  0.1135000135583575  449 ^ 1
  0.1132827525593783  5 ^ 5
  0.1131727366659431  457 ^ 1
  0.1130119349340791  461 ^ 1
  0.1122886528902697  37 ^ 2
  0.1121589678232448  13 ^ 3
  0.1112158783706281  509 ^ 1
  0.1108016106675626  521 ^ 1
  0.1101384080494621  541 ^ 1
  0.1096306885348829  557 ^ 1
  0.1092623336390804  569 ^ 1
   0.109184661243966  41 ^ 2
   0.109022393369416  577 ^ 1
  0.1085553762730775  593 ^ 1
  0.1083280294582822  601 ^ 1
  0.1079943548673958  613 ^ 1
  0.1078850289369187  617 ^ 1
  0.1072480305374002  641 ^ 1
  0.1069411297181192  653 ^ 1
  0.1067405999140516  661 ^ 1
  0.1064456832186591  673 ^ 1
  0.1063489015533347  677 ^ 1
  0.1057834942757523  701 ^ 1
  0.1056006150284899  709 ^ 1
  0.1050677386679089  733 ^ 1
  0.1045571277453435  757 ^ 1
  0.1044740746281937  761 ^ 1
  0.1043096601296809  769 ^ 1
  0.1042282850833419  773 ^ 1
   0.103751273435192  797 ^ 1
   0.103519712755121  809 ^ 1
  0.1032925703785417  821 ^ 1
  0.1031435227654369  829 ^ 1
  0.1027073457992096  853 ^ 1
  0.1026361963957853  857 ^ 1
  0.1022867948546604  877 ^ 1
  0.1022181522110647  881 ^ 1
  0.1021247598253814  53 ^ 2
   0.101539149197962  17 ^ 3
  0.1014246637053439  929 ^ 1
  0.1012975688001153  937 ^ 1
  0.1012345460491029  941 ^ 1
  0.1010475355097117  953 ^ 1
  0.1006824786789692  977 ^ 1
  0.1003869949333126  997 ^ 1
  0.1002133496204834  1009 ^ 1
  0.1001560585753007  1013 ^ 1
  0.1000423464158209  1021 ^ 1
 0.09987391378937605  1033 ^ 1
 0.09965321714259505  1049 ^ 1
 0.09949051958609365  1061 ^ 1
 0.09938336460724531  1069 ^ 1
 0.09906799126575533  1093 ^ 1
 0.09901629479181956  1097 ^ 1
 0.09886264801081832  1109 ^ 1
 0.09876139901991253  1117 ^ 1
 0.09863233986375369  61 ^ 2
 0.09861125988744147  1129 ^ 1
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  • $\begingroup$ Hi, I'm probably stupid but I don't see how highly composite numbers help with Pythagorean triples $a^2 + b^2 = c^2$. $\endgroup$
    – Labo
    Commented Nov 1, 2020 at 21:26
  • $\begingroup$ The number of right triangles with hypotenuse $n$ is related to the prime divisors of $n$ of the form $4k+1$ in much the same way that the highly composite numbers are related to their prime divisors. $\endgroup$ Commented Nov 2, 2020 at 6:31
  • $\begingroup$ @Gerry thanks. I put a second answer with better printout. The appeal to number of divisors is from Dickson's little 1929 Intro book, if I can find my copy I will add in the page and theorem numbers. $\endgroup$
    – Will Jagy
    Commented Nov 2, 2020 at 19:52
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Too big for a comment. These are the factors of the series elements. Perhaps you can find a pattern.

\begin{align*} 1&=5^0\\ 5&=5^1\\ 25&=5^2\\ 125&=5^3\\ 65&=5*13\\ 3125&=5^5\\ 15625&=5^6\\ 325&=5^2*13\\ 390625&=5^8\\ 1953125&=5^9\\ 1625&=5^3*13\\ 48828125&=5^{11}\\ 4225&=5^2*13^2\\ 1105&=5*13*17\\ 6103515625&=5^{14}\\  30517578125&=5^{15}\\  40625&=5^5*13\\  21125&=5^3*13^2\\  3814697265625&=5^{18}\\  203125&=5^6*13\\  95367431640625&=5^{20}\\  476837158203125&=5^{21}\\  5525&=5^2*13*17\\ 11920928955078125&=5^{23}\\ 274625&=5^3*13^3\\ \end{align*}

If you go here you can find this series taken to $1438$ terms. If you use WolframAlpha as I did, you can perhaps find a pattern in the elements with factors of $13$ and $17$ and $29$ such as

$$2005628955078125 = 5^{11}×13^2×17^2×29^2$$

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I worked out a program with better printing. These are the first 95 numbers with extreme efficiency in creating large numbers of representations. They get too long either written in decimal or factored; I stuck with factored. The number "reps" is the count of $$ a^2 + b^2 = n \; , \; \; a > b > 0 $$ The first is $$ 5 = 2^2 + 1^2 $$ The second is $$ 65 = 8^2 + 1^2 = 7^2 + 4^2 $$ The third is $$ 325 = 18^2 + 1^2 = 17^2 + 6^2 = 15^2+ 10^2 $$ The fourth is $$ 5525 = 74^2 + 7^2 = 73^2 + 14^2 = 71^2+ 22^2 = 70^2 + 25^2 = 62^2 + 41^2 = 55^2+ 50^2 $$ The relevant theorem is in Dickson's little book, 1929, Introduction to the Theory of Numbers in which he says the exact number of representations of a number using the number of divisors of form $1 \pmod 4$ and subtracting off the number of divisors of form $3 \pmod 4,$ then multiplying by $4$ for all of them (including negatives, reversing order). For us, the numbers are never squares, so zero is not involved. The result is that taking half of Dickson's number (he calls it $E$ for excess) is correct here. Found it, page 80, section 51, Theorem 65.

enter image description here

 ( 0.4306765580733931 , 5 , 1 )   reps  1 prod   5
 ( 0.2702381544273197 , 13 , 1 )   reps  2 prod   5 13
 ( 0.2519296364125923 , 5 , 2 )   reps  3 prod   5^2 13
 ( 0.244650542118226 , 17 , 1 )   reps  6 prod   5^2 13 17
 ( 0.2058468324604344 , 29 , 1 )   reps  12 prod   5^2 13 17 29
 ( 0.1919587200065601 , 37 , 1 )   reps  24 prod   5^2 13 17 29 37
 ( 0.1866524112389434 , 41 , 1 )   reps  48 prod   5^2 13 17 29 37 41
 ( 0.1787469216608008 , 5 , 3 )   reps  64 prod   5^3 13 17 29 37 41
 ( 0.1745834300480449 , 53 , 1 )   reps  128 prod   5^3 13 17 29 37 41 53
 ( 0.1686130986895011 , 61 , 1 )   reps  256 prod   5^3 13 17 29 37 41 53 61
 ( 0.1615554674429964 , 73 , 1 )   reps  512 prod   5^3 13 17 29 37 41 53 61 73
 ( 0.158079186604075 , 13 , 2 )   reps  768 prod   5^3 13^2 17 29 37 41 53 61 73
 ( 0.1544226628011101 , 89 , 1 )   reps  1536 prod   5^3 13^2 17 29 37 41 53 61 73 89
 ( 0.1515171524096389 , 97 , 1 )   reps  3072 prod   5^3 13^2 17 29 37 41 53 61 73 89 97
 ( 0.150190483223688 , 101 , 1 )   reps  6144 prod   5^3 13^2 17 29 37 41 53 61 73 89 97 101
 ( 0.1477501131786861 , 109 , 1 )   reps  12288 prod   5^3 13^2 17 29 37 41 53 61 73 89 97 101 109
 ( 0.1466237184553111 , 113 , 1 )   reps  24576 prod   5^3 13^2 17 29 37 41 53 61 73 89 97 101 109 113
 ( 0.1431113929202641 , 17 , 2 )   reps  36864 prod   5^3 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113
 ( 0.1408841194731412 , 137 , 1 )   reps  73728 prod   5^3 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137
 ( 0.1386468838532138 , 5 , 4 )   reps  92160 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137
 ( 0.1385201075671774 , 149 , 1 )   reps  184320 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149
 ( 0.1370873187823978 , 157 , 1 )   reps  368640 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157
 ( 0.1345057169479844 , 173 , 1 )   reps  737280 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173
 ( 0.1333360731748201 , 181 , 1 )   reps  1474560 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181
 ( 0.1317096685686114 , 193 , 1 )   reps  2949120 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193
 ( 0.1311982683517524 , 197 , 1 )   reps  5898240 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197
 ( 0.1275639755045533 , 229 , 1 )   reps  11796480 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229
 ( 0.1271587398372755 , 233 , 1 )   reps  23592960 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233
 ( 0.1263760881150453 , 241 , 1 )   reps  47185920 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241
 ( 0.1249121781636255 , 257 , 1 )   reps  94371840 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257
 ( 0.1238932866290727 , 269 , 1 )   reps  188743680 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269
 ( 0.1232476925609442 , 277 , 1 )   reps  377487360 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277
 ( 0.1229342993142804 , 281 , 1 )   reps  754974720 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281
 ( 0.1220292459879827 , 293 , 1 )   reps  1509949440 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293
 ( 0.1206269875212659 , 313 , 1 )   reps  3019898880 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313
 ( 0.1204126778815846 , 29 , 2 )   reps  4529848320 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313
 ( 0.1203610007213705 , 317 , 1 )   reps  9059696640 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317
 ( 0.1190957566850307 , 337 , 1 )   reps  18119393280 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337
 ( 0.1183840591148901 , 349 , 1 )   reps  36238786560 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349
 ( 0.1181540875758708 , 353 , 1 )   reps  72477573120 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353
 ( 0.1170544627527939 , 373 , 1 )   reps  144955146240 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373
 ( 0.1162300592682939 , 389 , 1 )   reps  289910292480 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389
 ( 0.1158346526511383 , 397 , 1 )   reps  579820584960 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397
 ( 0.115640914438019 , 401 , 1 )   reps  1159641169920 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401
 ( 0.1152610595238196 , 409 , 1 )   reps  2319282339840 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409
 ( 0.1147094651682719 , 421 , 1 )   reps  4638564679680 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421
 ( 0.1141784098763921 , 433 , 1 )   reps  9277129359360 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433
 ( 0.1135000135583575 , 449 , 1 )   reps  18554258718720 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449
 ( 0.1132827525593783 , 5 , 5 )   reps  22265110462464 prod   5^5 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449
 ( 0.1131727366659431 , 457 , 1 )   reps  44530220924928 prod   5^5 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457
 ( 0.1130119349340791 , 461 , 1 )   reps  89060441849856 prod   5^5 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461
 ( 0.1122886528902697 , 37 , 2 )   reps  133590662774784 prod   5^5 13^2 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461
 ( 0.1121589678232448 , 13 , 3 )   reps  178120883699712 prod   5^5 13^3 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461
 ( 0.1112158783706281 , 509 , 1 )   reps  356241767399424 prod   5^5 13^3 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509
 ( 0.1108016106675626 , 521 , 1 )   reps  712483534798848 prod   5^5 13^3 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521
 ( 0.1101384080494621 , 541 , 1 )   reps  1424967069597696 prod   5^5 13^3 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541
 ( 0.1096306885348829 , 557 , 1 )   reps  2849934139195392 prod   5^5 13^3 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557
 ( 0.1092623336390804 , 569 , 1 )   reps  5699868278390784 prod   5^5 13^3 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569
 ( 0.109184661243966 , 41 , 2 )   reps  8549802417586176 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569
 ( 0.109022393369416 , 577 , 1 )   reps  17099604835172352 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577
 ( 0.1085553762730775 , 593 , 1 )   reps  34199209670344704 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593
 ( 0.1083280294582822 , 601 , 1 )   reps  68398419340689408 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601
 ( 0.1079943548673958 , 613 , 1 )   reps  136796838681378816 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613
 ( 0.1078850289369187 , 617 , 1 )   reps  273593677362757632 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617
 ( 0.1072480305374002 , 641 , 1 )   reps  547187354725515264 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641
 ( 0.1069411297181192 , 653 , 1 )   reps  1094374709451030528 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653
 ( 0.1067405999140516 , 661 , 1 )   reps  2188749418902061056 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661
 ( 0.1064456832186591 , 673 , 1 )   reps  4377498837804122112 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673
 ( 0.1063489015533347 , 677 , 1 )   reps  8754997675608244224 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677
 ( 0.1057834942757523 , 701 , 1 )   reps  17509995351216488448 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701
 ( 0.1056006150284899 , 709 , 1 )   reps  35019990702432976896 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709
 ( 0.1050677386679089 , 733 , 1 )   reps  70039981404865953792 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733
 ( 0.1045571277453435 , 757 , 1 )   reps  140079962809731907584 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757
 ( 0.1044740746281937 , 761 , 1 )   reps  280159925619463815168 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761
 ( 0.1043096601296809 , 769 , 1 )   reps  560319851238927630336 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769
 ( 0.1042282850833419 , 773 , 1 )   reps  1120639702477855260672 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773
 ( 0.103751273435192 , 797 , 1 )   reps  2241279404955710521344 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797
 ( 0.103519712755121 , 809 , 1 )   reps  4482558809911421042688 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809
 ( 0.1032925703785417 , 821 , 1 )   reps  8965117619822842085376 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821
 ( 0.1031435227654369 , 829 , 1 )   reps  17930235239645684170752 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829
 ( 0.1027073457992096 , 853 , 1 )   reps  35860470479291368341504 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853
 ( 0.1026361963957853 , 857 , 1 )   reps  71720940958582736683008 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857
 ( 0.1022867948546604 , 877 , 1 )   reps  143441881917165473366016 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877
 ( 0.1022181522110647 , 881 , 1 )   reps  286883763834330946732032 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881
 ( 0.1021247598253814 , 53 , 2 )   reps  430325645751496420098048 prod   5^5 13^3 17^2 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881
 ( 0.101539149197962 , 17 , 3 )   reps  573767527668661893464064 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881
 ( 0.1014246637053439 , 929 , 1 )   reps  1147535055337323786928128 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929
 ( 0.1012975688001153 , 937 , 1 )   reps  2295070110674647573856256 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937
 ( 0.1012345460491029 , 941 , 1 )   reps  4590140221349295147712512 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941
 ( 0.1010475355097117 , 953 , 1 )   reps  9180280442698590295425024 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941 953
 ( 0.1006824786789692 , 977 , 1 )   reps  18360560885397180590850048 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941 953 977
 ( 0.1003869949333126 , 997 , 1 )   reps  36721121770794361181700096 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941 953 977 997
 ( 0.1002133496204834 , 1009 , 1 )   reps  73442243541588722363400192 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941 953 977 997 1009
 ( 0.1001560585753007 , 1013 , 1 )   reps  146884487083177444726800384 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941 953 977 997 1009 1013
 ( 0.1000423464158209 , 1021 , 1 )   reps  293768974166354889453600768 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941 953 977 997 1009 1013 1021 
$\endgroup$

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