%I M4682 N2001 #35 May 13 2020 07:03:37
%S 1,0,1,10,-17,406,-1437,20476,-44907,1068404,-5112483,230851094,
%T -1942311373,31916614874,-27260241361,3826126294680,-37957167335671,
%U 2169009251237640,-25847377785179111,858747698098918338,-5611513985867158697,154094365406716365118
%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="/A002744/b002744.txt">Table of n, a(n) for n = 1..451</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)*A000005(k)*(k-1)!*binomial(n, k). - _Vladeta Jovovic_, Feb 09 2003
%F E.g.f.: -exp(-x) * log(Product_{k>=1} (1 - x^k)^(1/k)). - _Ilya Gutkovskiy_, Dec 11 2019
%F a(p) == -2 (mod p) for prime p. The pseudoprimes of this congruence are 4, 6, 20, 42, 1806, ... - _Amiram Eldar_, May 13 2020
%t 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 *)
%o (PARI) a(n) = sum(k=1, n, (-1)^(n-k)*numdiv(k)*(k-1)!*binomial(n, k)); \\ _Michel Marcus_, May 13 2020
%Y Cf. A000005, A002746, A318249.
%K sign
%O 1,4
%A _N. J. A. Sloane_
%E Corrected and extended by _Jeffrey Shallit_
%E More terms from _Vladeta Jovovic_, Feb 09 2003