A002743 Sum of logarithmic numbers. (Formerly M2132 N0845)

1, 1, 2, 24, -11, 1085, -2542, 64344, -56415, 4275137, -10660486, 945005248, -6010194555, 147121931021, 88135620922, 23131070531152, -120142133444319, 12007306976370081, -103897545509370542, 4923827766711915784, -19471338470911446283, 1203786171449486366205

Offset

1,3

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..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

Formula

a(n) = Sum_{k=1..n} (-1)^(n-k)*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 6, 42, 1806, ... - Amiram Eldar, May 13 2020

Mathematica

a[n_] := n! * Sum[(-1)^k * DivisorSigma[1, 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)*sigma(k)*(k-1)!*binomial(n, k)); \\ Michel Marcus, May 13 2020

Crossrefs

Cf. A000203, A002745.

Sequence in context: A066585 A278563 A075267 * A220773 A290772 A342545

Adjacent sequences: A002740 A002741 A002742 * A002744 A002745 A002746

Keyword

sign

Author

N. J. A. Sloane

Extensions

More terms from Jeffrey Shallit

Status

approved