A002744 Sum of logarithmic numbers.

1, 0, 1, 10, -17, 406, -1437, 20476, -44907, 1068404, -5112483, 230851094, -1942311373, 31916614874, -27260241361, 3826126294680, -37957167335671, 2169009251237640, -25847377785179111, 858747698098918338, -5611513985867158697, 154094365406716365118

Offset: 1

Author

N. J. A. Sloane

Keywords

• sign

References

• J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.
• 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..451
• 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. A000005, A002746, A318249.

Programs

Mathematica

a[n_] := n! * Sum[(-1)^k * DivisorSigma[0, n - k]/k!/(n - k), {k, 0, n - 1}]; Array[a, 22] (* Amiram Eldar, May 13 2020 *)

PARI

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

Formula

a(n) = Sum_{k=1..n} (-1)^(n-k)*A000005(k)*(k-1)!*binomial(n, k). - Vladeta Jovovic, Feb 09 2003

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

a(p) == -2 (mod p) for prime p. The pseudoprimes of this congruence are 4, 6, 20, 42, 1806, ... - Amiram Eldar, May 13 2020

Extensions

Corrected and extended by Jeffrey Shallit

More terms from Vladeta Jovovic, Feb 09 2003