A076751 a(n) is the smallest composite k such that Sum_{composites j = 4, ..., k} 1/j exceeds n.

16, 63, 216, 715, 2279, 7102, 21722, 65558, 195759, 579465, 1703072, 4975222, 14459492, 41837580, 120585504, 346372172, 991915208, 2832896772, 8071045528, 22944211170

Offset

1,1

Comments

These partial sums, like the harmonic sequence (A004080), can never be integers.

Links

Table of n, a(n) for n=1..20.

Formula

Limit_{n->oo} a(n+1)/a(n) = e.

a(n) = A002808(A074631(n)). - Amiram Eldar, Jul 17 2024

Example

a(1) = 1 because 1/4 + 1/6 + 1/8 + 1/9 + 1/10 + 1/12 + 1/14 + 1/15 = 0.97420... < 1 but 1/4 + 1/6 + 1/8 + 1/9 + 1/10 + 1/12 + 1/14 + 1/15 + 1/16 = 1.03670... > 1.

Mathematica

NextComposite[n_] := Block[{k = n + 1}, While[ PrimeQ[k], k++ ]; k]; k = 4; s = 0; Do[ While[s = s + 1/k; s < n, k = NextComposite[k]]; Print[k]; k = NextComposite[k], {n, 1, 17}]

Prog

(PARI) lista(cmax) = {my(n = 1, s = 0); forcomposite(c = 1, cmax, s += 1/c; if(s > n, print1(c, ", "); n++)); } \\ Amiram Eldar, Jul 17 2024

Crossrefs

Cf. A002808, A004080, A016088, A074631.

Sequence in context: A143860 A100176 A060091 * A215969 A333663 A155985

Adjacent sequences: A076748 A076749 A076750 * A076752 A076753 A076754

Keyword

nonn,hard,more

Author

Jack Brennen, Nov 12 2002

Extensions

Edited and extended by Robert G. Wilson v, Nov 14 2002

Name edited and a(18) added by Jon E. Schoenfield, Feb 01 2020

a(19)-a(20) from Amiram Eldar, Jul 17 2024

Status

approved