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I was browsing OEIS and came across the largely composite numbers, A067128, defined as the natural numbers that have at least as many divisors as all smaller natural numbers. (They are of course related to the highly composite numbers.)

A comment on the OEIS page asks whether the largely composite numbers are the same as A034287, the numbers $n$ such that the product of the divisors of $n$ is larger than for all smaller natural numbers. In reply, another comment says that the two sequences are the same for all terms less than $10^{150}$, of which there are 105834.

My questions are:

  1. Are these two sequences the same, or do they differ at some point after the 105834th term?

  2. If they do differ, is there a nice way to see why the two sequences should be the same for such a large range of initial values?

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  • $\begingroup$ @KinnisalMountainChicken I don't know if the 105835th term is different, but the comment on OEIS implies that it is. I'd originally intended to ask for proof, but my text was ambiguous and somebody disambiguated it to remove that implication. I've restored the original question, asking whether it is true as well as asking for the reason why. $\endgroup$ – Nathaniel Mar 7 '16 at 0:23
  • $\begingroup$ Nathaniel, the wording of the comment reflects the fact that these sequences are generated by computers; the person who posted that stopped the computer at a round number, $10^{150}.$ The count of that list turned out to be 105834, but the person making that run did not check any higher, and has no idea what happens next. This is not necessarily general mathematics use of English, but it is the way they say things on OEIS. $\endgroup$ – Will Jagy Mar 7 '16 at 0:35
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    $\begingroup$ @WillJagy thanks, I began to suspect that after reading your answer and the comments this morning - I hadn't thought of it when I posted. So perhaps the sequences differ at some point or perhaps they are the same - it will be interesting to find out. I'll edit the question to remove the implication that they differ after the 105834th term. $\endgroup$ – Nathaniel Mar 7 '16 at 0:41
  • $\begingroup$ @Nathaniel: The old title was catchier and the present one is a bit generic. :) Maybe you can have something like, "Do these two sequences differ infinitely many times after the 105834th term?" $\endgroup$ – Tito Piezas III Mar 7 '16 at 2:55
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Begun actual work on the thing. The product of divisors has, of course, the same prime factors as the original number. What I did not know is that, if the original exponent is $a$ and the number of divisors of the number is $d(n),$ then the new exponent of that prime (in the product of divisors) is $$ a \, d(n) / 2. $$ This gives the first hint of how a large number of divisors tends to give a large product of divisors, in a tightly controlled manner. Put another way, if the original number is $n$ and the product of all divisors is $P,$ then $$ P = n^{d(n)/2} $$ Therefore, if $n$ has at least as many divisors as all smaller numbers, then $P$ is guaranteed strictly larger than all previous values for $P.$ So, we have that A067128 is contained in A034287, maybe strictly, or maybe the sequences are equal.

THEOREM a largely composite number has a product of divisors that is strictly larger than such products for all smaller numbers.

Approaches for the other direction: if the assumption is that $n$ sets a new record for product of divisors, we are saying that, for all $m < n,$ $$ d(n) > \left( \frac{\log m}{\log n} \right) d(m). $$ We do have explicit upper bounds on the size of $d(n)$ due to Nicolas and Robin; the important thing is how very small these bounds are. It is possible that numbers setting new divisor product records are so frequent that, when $m$ is the previous element in that list, that $ \left( \frac{\log m}{\log n} \right) d(m) > d(m) - 1. $ That would do it; maybe it is true. I will, at least, experiment with that. OH, WELL. The conjectured inequality does not appear to be true, or even true for sufficently large numbers. On the other hand, we appear to get the promising $ \left( \frac{\log m}{\log n} \right) d(m) > d(m) - 3. $ Worth playing with this computer conjecture, because $d(m)$ is even unless $m$ itself is a square. NOPE. The $3$ does not hold up either. Here are the smallest numbers where the difference exceeds 2.0. The way this is going, I think either finding a number on one list but not the other, or a proof the lists are the same, would be a fair amount of effort.

7560 =  ( 3,  3,  1,  1  )  prod    =  ( 96,  96,  32,  32,  )   number of divisors  64 prev  60 57.2759  diff  2.7241
131040 =  ( 5,  2,  1,  1,  1  )  prod    =  ( 360,  144,  72,  72,  72,  )   number of divisors  144 prev  144 141.958  diff  2.04152
196560 =  ( 4,  3,  1,  1,  1  )  prod    =  ( 320,  240,  80,  80,  80,  )   number of divisors  160 prev  160 157.807  diff  2.1929
262080 =  ( 6,  2,  1,  1,  1  )  prod    =  ( 504,  168,  84,  84,  84,  )   number of divisors  168 prev  168 165.751  diff  2.24945
327600 =  ( 4,  2,  2,  1,  1  )  prod    =  ( 360,  180,  180,  90,  90,  )   number of divisors  180 prev  180 177.632  diff  2.36778
655200 =  ( 5,  2,  2,  1,  1  )  prod    =  ( 540,  216,  216,  108,  108,  )   number of divisors  216 prev  216 213.306  diff  2.69428
831600 =  ( 4,  3,  2,  1,  1  )  prod    =  ( 480,  360,  240,  120,  120,  )   number of divisors  240 prev  240 237.48  diff  2.51955
942480 =  ( 4,  2,  1,  1,  1,  1  )  prod    =  ( 480,  240,  120,  120,  120,  120,  )   number of divisors  240 prev  240 237.816  diff  2.18367
1330560 =  ( 7,  3,  1,  1,  1  )  prod    =  ( 896,  384,  128,  128,  128,  )   number of divisors  256 prev  256 252.23  diff  3.76961
1663200 =  ( 5,  3,  2,  1,  1  )  prod    =  ( 720,  432,  288,  144,  144,  )   number of divisors  288 prev  288 285.123  diff  2.87715

Sample, just the exponents, not the primes themselves:

2 =  (  1 )  prod    =  (  1 )   number of divisors  2
3 =  (  1 )  prod    =  (  1 )   number of divisors  2
4 =  ( 2,  )  prod    =  ( 3,  )   number of divisors  3
6 =  ( 1,  1  )  prod    =  ( 2,  2,  )   number of divisors  4
8 =  ( 3,  )  prod    =  ( 6,  )   number of divisors  4
10 =  ( 1,  1  )  prod    =  ( 2,  2,  )   number of divisors  4
12 =  ( 2,  1  )  prod    =  ( 6,  3,  )   number of divisors  6
18 =  ( 1,  2,  )  prod    =  ( 3,  6,  )   number of divisors  6
20 =  ( 2,  1  )  prod    =  ( 6,  3,  )   number of divisors  6
24 =  ( 3,  1  )  prod    =  ( 12,  4,  )   number of divisors  8
30 =  ( 1,  1,  1  )  prod    =  ( 4,  4,  4,  )   number of divisors  8
36 =  ( 2,  2,  )  prod    =  ( 9,  9,  )   number of divisors  9
48 =  ( 4,  1  )  prod    =  ( 20,  5,  )   number of divisors  10
60 =  ( 2,  1,  1  )  prod    =  ( 12,  6,  6,  )   number of divisors  12
72 =  ( 3,  2,  )  prod    =  ( 18,  12,  )   number of divisors  12
84 =  ( 2,  1,  1  )  prod    =  ( 12,  6,  6,  )   number of divisors  12
90 =  ( 1,  2,  1  )  prod    =  ( 6,  12,  6,  )   number of divisors  12
96 =  ( 5,  1  )  prod    =  ( 30,  6,  )   number of divisors  12
108 =  ( 2,  3,  )  prod    =  ( 12,  18,  )   number of divisors  12
120 =  ( 3,  1,  1  )  prod    =  ( 24,  8,  8,  )   number of divisors  16
168 =  ( 3,  1,  1  )  prod    =  ( 24,  8,  8,  )   number of divisors  16
180 =  ( 2,  2,  1  )  prod    =  ( 18,  18,  9,  )   number of divisors  18
240 =  ( 4,  1,  1  )  prod    =  ( 40,  10,  10,  )   number of divisors  20
336 =  ( 4,  1,  1  )  prod    =  ( 40,  10,  10,  )   number of divisors  20
360 =  ( 3,  2,  1  )  prod    =  ( 36,  24,  12,  )   number of divisors  24
420 =  ( 2,  1,  1,  1  )  prod    =  ( 24,  12,  12,  12,  )   number of divisors  24
480 =  ( 5,  1,  1  )  prod    =  ( 60,  12,  12,  )   number of divisors  24
504 =  ( 3,  2,  1  )  prod    =  ( 36,  24,  12,  )   number of divisors  24
540 =  ( 2,  3,  1  )  prod    =  ( 24,  36,  12,  )   number of divisors  24
600 =  ( 3,  1,  2,  )  prod    =  ( 36,  12,  24,  )   number of divisors  24
630 =  ( 1,  2,  1,  1  )  prod    =  ( 12,  24,  12,  12,  )   number of divisors  24
660 =  ( 2,  1,  1,  1  )  prod    =  ( 24,  12,  12,  12,  )   number of divisors  24
672 =  ( 5,  1,  1  )  prod    =  ( 60,  12,  12,  )   number of divisors  24
720 =  ( 4,  2,  1  )  prod    =  ( 60,  30,  15,  )   number of divisors  30
840 =  ( 3,  1,  1,  1  )  prod    =  ( 48,  16,  16,  16,  )   number of divisors  32
1080 =  ( 3,  3,  1  )  prod    =  ( 48,  48,  16,  )   number of divisors  32
1260 =  ( 2,  2,  1,  1  )  prod    =  ( 36,  36,  18,  18,  )   number of divisors  36
1440 =  ( 5,  2,  1  )  prod    =  ( 90,  36,  18,  )   number of divisors  36
1680 =  ( 4,  1,  1,  1  )  prod    =  ( 80,  20,  20,  20,  )   number of divisors  40
2160 =  ( 4,  3,  1  )  prod    =  ( 80,  60,  20,  )   number of divisors  40
2520 =  ( 3,  2,  1,  1  )  prod    =  ( 72,  48,  24,  24,  )   number of divisors  48
3360 =  ( 5,  1,  1,  1  )  prod    =  ( 120,  24,  24,  24,  )   number of divisors  48
3780 =  ( 2,  3,  1,  1  )  prod    =  ( 48,  72,  24,  24,  )   number of divisors  48
3960 =  ( 3,  2,  1,  1  )  prod    =  ( 72,  48,  24,  24,  )   number of divisors  48
4200 =  ( 3,  1,  2,  1  )  prod    =  ( 72,  24,  48,  24,  )   number of divisors  48
4320 =  ( 5,  3,  1  )  prod    =  ( 120,  72,  24,  )   number of divisors  48
4620 =  ( 2,  1,  1,  1,  1  )  prod    =  ( 48,  24,  24,  24,  24,  )   number of divisors  48
4680 =  ( 3,  2,  1,  1  )  prod    =  ( 72,  48,  24,  24,  )   number of divisors  48
5040 =  ( 4,  2,  1,  1  )  prod    =  ( 120,  60,  30,  30,  )   number of divisors  60
7560 =  ( 3,  3,  1,  1  )  prod    =  ( 96,  96,  32,  32,  )   number of divisors  64
9240 =  ( 3,  1,  1,  1,  1  )  prod    =  ( 96,  32,  32,  32,  32,  )   number of divisors  64
10080 =  ( 5,  2,  1,  1  )  prod    =  ( 180,  72,  36,  36,  )   number of divisors  72
12600 =  ( 3,  2,  2,  1  )  prod    =  ( 108,  72,  72,  36,  )   number of divisors  72
13860 =  ( 2,  2,  1,  1,  1  )  prod    =  ( 72,  72,  36,  36,  36,  )   number of divisors  72
15120 =  ( 4,  3,  1,  1  )  prod    =  ( 160,  120,  40,  40,  )   number of divisors  80
18480 =  ( 4,  1,  1,  1,  1  )  prod    =  ( 160,  40,  40,  40,  40,  )   number of divisors  80
20160 =  ( 6,  2,  1,  1  )  prod    =  ( 252,  84,  42,  42,  )   number of divisors  84
25200 =  ( 4,  2,  2,  1  )  prod    =  ( 180,  90,  90,  45,  )   number of divisors  90
27720 =  ( 3,  2,  1,  1,  1  )  prod    =  ( 144,  96,  48,  48,  48,  )   number of divisors  96
30240 =  ( 5,  3,  1,  1  )  prod    =  ( 240,  144,  48,  48,  )   number of divisors  96
32760 =  ( 3,  2,  1,  1,  1  )  prod    =  ( 144,  96,  48,  48,  48,  )   number of divisors  96
36960 =  ( 5,  1,  1,  1,  1  )  prod    =  ( 240,  48,  48,  48,  48,  )   number of divisors  96
37800 =  ( 3,  3,  2,  1  )  prod    =  ( 144,  144,  96,  48,  )   number of divisors  96
40320 =  ( 7,  2,  1,  1  )  prod    =  ( 336,  96,  48,  48,  )   number of divisors  96
41580 =  ( 2,  3,  1,  1,  1  )  prod    =  ( 96,  144,  48,  48,  48,  )   number of divisors  96
42840 =  ( 3,  2,  1,  1,  1  )  prod    =  ( 144,  96,  48,  48,  48,  )   number of divisors  96
43680 =  ( 5,  1,  1,  1,  1  )  prod    =  ( 240,  48,  48,  48,  48,  )   number of divisors  96

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