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Sum of logarithmic numbers.
(Formerly M3909 N1604)
7

%I M3909 N1604 #38 Oct 20 2023 09:37:23

%S 1,5,20,96,469,3145,20684,173544,1557105,16215253,159346604,

%T 2230085528,26985045333,368730610729,5628888393652,97987283458928,

%U 1475486672174337,29097611462122437,505383110562327268,10970329921706735216

%N Sum of logarithmic numbers.

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

%D Jeffrey Shallit, personal communication.

%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="/A002745/b002745.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} 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 30, 858, 1722, ... - _Amiram Eldar_, May 13 2020

%t Table[Sum[Binomial[n,k] * DivisorSigma[1,k] * (k-1)!, {k, 1, n}], {n, 1, 20}] (* _Vaclav Kotesovec_, Dec 16 2019 *)

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

%Y Cf. A000203, A002743.

%K nonn

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Vladeta Jovovic_, Feb 09 2003