A002744 Sum of logarithmic numbers. (Formerly M4682 N2001)

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

Offset

1,4

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

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

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 *)

Prog

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

Crossrefs

Cf. A000005, A002746, A318249.

Sequence in context: A219879 A177185 A241281 * A320930 A337325 A156382

Adjacent sequences: A002741 A002742 A002743 * A002745 A002746 A002747

Keyword

sign

Author

N. J. A. Sloane

Extensions

Corrected and extended by Jeffrey Shallit

More terms from Vladeta Jovovic, Feb 09 2003

Status

approved