A002743 Sum of logarithmic numbers.

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

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..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, A002745.

Programs

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 *)

PARI

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

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

Extensions

More terms from Jeffrey Shallit