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A000441 a(n) = Sum_{k=1..n-1} k*sigma(k)*sigma(n-k).
(Formerly M4613 N1968)
4

%I M4613 N1968

%S 0,1,9,34,95,210,406,740,1161,1920,2695,4116,5369,7868,9690,13640,

%T 16116,22419,25365,34160,38640,50622,55154,73320,77225,100100,107730,

%U 135576,141085,182340,184760,233616,243408,297738,301420,385110,377511,467210,478842

%N a(n) = Sum_{k=1..n-1} k*sigma(k)*sigma(n-k).

%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).

%D J. Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39.

%H Vaclav Kotesovec, <a href="/A000441/b000441.txt">Table of n, a(n) for n = 1..1000</a>

%H J. Touchard, <a href="/A000385/a000385.pdf">On prime numbers and perfect numbers</a>, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]

%F a(n) = Sum_{k=1..n-1} k * sigma(k) * sigma(n-k). Convolution of A000203 with A064987. - _Sean A. Irvine_, Nov 14 2010

%F G.f.: x*f(x)*f'(x), where f(x) = Sum_{k>=1} k*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Apr 28 2018

%p S:=(n,e)->add(k^e*sigma(k)*sigma(n-k),k=1..n-1);

%p f:=e->[seq(S(n,e),n=1..30)];f(1); # _N. J. A. Sloane_, Jul 03 2015

%t a[n_] := Sum[k*DivisorSigma[1, k]*DivisorSigma[1, n-k], {k, 1, n-1}]; Array[a, 40] (* _Jean-Fran├žois Alcover_, Feb 08 2016 *)

%o (PARI) a(n) = sum(k=1, n-1, k*sigma(k)*sigma(n-k)); \\ _Michel Marcus_, Feb 02 2014

%Y Cf. A000441, A000499.

%K nonn

%O 1,3

%A _N. J. A. Sloane_

%E More terms from _Sean A. Irvine_, Nov 14 2010

%E a(1)=0 prepended by _Michel Marcus_, Feb 02 2014

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Last modified September 20 06:51 EDT 2020. Contains 337264 sequences. (Running on oeis4.)