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Sum of logarithmic numbers.
(Formerly M2132 N0845)
5

%I M2132 N0845 #34 Oct 20 2023 09:36:27

%S 1,1,2,24,-11,1085,-2542,64344,-56415,4275137,-10660486,945005248,

%T -6010194555,147121931021,88135620922,23131070531152,-120142133444319,

%U 12007306976370081,-103897545509370542,4923827766711915784,-19471338470911446283,1203786171449486366205

%N Sum of logarithmic numbers.

%D J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Amiram Eldar, <a href="/A002743/b002743.txt">Table of n, a(n) for n = 1..449</a>

%H J. M. Gandhi, <a href="/A002741/a002741.pdf">On logarithmic numbers</a>, Math. Student, 31 (1963), 73-83. [Annotated scanned copy]

%H J. M. Gandhi, <a href="https://doi.org/10.1080/00029890.1966.11970871">Logarithmic Numbers and the Functions d(n) and sigma(n)</a>, The American Mathematical Monthly, Vol. 73, No. 9 (1966), pp. 959-964, <a href="https://www.jstor.org/stable/2314495">alternative link</a>.

%H <a href="/index/Lo#logarithmic">Index entries for sequences related to logarithmic numbers</a>

%F a(n) = Sum_{k=1..n} (-1)^(n-k)*A000203(k)*(k-1)!*binomial(n, k). - _Vladeta Jovovic_, Feb 09 2003

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

%F a(p) == -1 (mod p) for prime p. The pseudoprimes of this congruence are 6, 42, 1806, ... - _Amiram Eldar_, May 13 2020

%t 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 *)

%o (PARI) a(n) = sum(k=1, n, (-1)^(n-k)*sigma(k)*(k-1)!*binomial(n, k)); \\ _Michel Marcus_, May 13 2020

%Y Cf. A000203, A002745.

%K sign

%O 1,3

%A _N. J. A. Sloane_

%E More terms from _Jeffrey Shallit_