A002745 Sum of logarithmic numbers.

1, 5, 20, 96, 469, 3145, 20684, 173544, 1557105, 16215253, 159346604, 2230085528, 26985045333, 368730610729, 5628888393652, 97987283458928, 1475486672174337, 29097611462122437, 505383110562327268, 10970329921706735216

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.
• Jeffrey Shallit, personal communication.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..449
• J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. [Annotated scanned copy]
• J. M. Gandhi, Logarithmic Numbers and the Functions d(n) and sigma(n), The American Mathematical Monthly, Vol. 73, No. 9 (1966), pp. 959-964, alternative link.
• Index entries for sequences related to logarithmic numbers

Crossrefs

Cf. A000203, A002743.

Programs

Mathematica

Table[Sum[Binomial[n,k] * DivisorSigma[1,k] * (k-1)!, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Dec 16 2019 *)

PARI

a(n) = sum(k=1, n, sigma(k)*(k-1)!*binomial(n, k)); \\ Michel Marcus, May 13 2020

Formula

a(n) = Sum_{k=1..n} A000203(k)*(k-1)!*binomial(n, k). - Vladeta Jovovic, Feb 09 2003

E.g.f.: exp(x) * Sum_{k>=1} x^k / (k*(1 - x^k)). - Ilya Gutkovskiy, Dec 11 2019

a(p) == -1 (mod p) for prime p. The pseudoprimes of this congruence are 30, 858, 1722, ... - Amiram Eldar, May 13 2020

Extensions

More terms from Vladeta Jovovic, Feb 09 2003