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