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%I M3468 N1411 #36 Oct 20 2023 09:38:10
%S 1,4,13,50,203,1154,6627,49356,403293,3858376,33929377,460614670,
%T 5168544119,64518640406,946910125319,16124114481720,221243980745433,
%U 4261440137319852,68524390012831189,1477309421907315082
%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="/A002746/b002746.txt">Table of n, a(n) for n = 1..450</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} 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, 12, 30, 380, 858, 1722 ... - _Amiram Eldar_, May 13 2020
%t Table[Sum[Binomial[n,k] * DivisorSigma[0,k] * (k-1)!, {k, 1, n}], {n, 1, 20}] (* _Vaclav Kotesovec_, Dec 16 2019 *)
%o (PARI) a(n) = sum(k=1, n, numdiv(k)*(k-1)!*binomial(n, k)); \\ _Michel Marcus_, May 13 2020
%Y Cf. A000005, A002744, A318249.
%K nonn
%O 1,2
%A _N. J. A. Sloane_
%E Corrected and extended by _Jeffrey Shallit_
%E More terms from _Vladeta Jovovic_, Feb 09 2003
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