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A000441
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a(n) = Sum_{k=1..n-1} k*sigma(k)*sigma(n-k).
(Formerly M4613 N1968)
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9
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0, 1, 9, 34, 95, 210, 406, 740, 1161, 1920, 2695, 4116, 5369, 7868, 9690, 13640, 16116, 22419, 25365, 34160, 38640, 50622, 55154, 73320, 77225, 100100, 107730, 135576, 141085, 182340, 184760, 233616, 243408, 297738, 301420, 385110, 377511, 467210, 478842
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OFFSET
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1,3
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COMMENTS
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Apart from initial zero this is the convolution of A340793 and A143128. - Omar E. Pol, Feb 16 2021
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39.
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LINKS
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Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
J. Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]
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FORMULA
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Convolution of A000203 with A064987. - Sean A. Irvine, Nov 14 2010
G.f.: x*f(x)*f'(x), where f(x) = Sum_{k>=1} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 28 2018
a(n) = (n/24 - n^2/4)*sigma_1(n) + (5*n/24)*sigma_3(n). - Ridouane Oudra, Sep 17 2020
Sum_{k=1..n} a(k) ~ Pi^4 * n^5 / 2160. - Vaclav Kotesovec, May 09 2022
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MAPLE
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S:=(n, e)->add(k^e*sigma(k)*sigma(n-k), k=1..n-1);
f:=e->[seq(S(n, e), n=1..30)]; f(1); # N. J. A. Sloane, Jul 03 2015
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MATHEMATICA
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a[n_] := Sum[k*DivisorSigma[1, k]*DivisorSigma[1, n-k], {k, 1, n-1}]; Array[a, 40] (* Jean-François Alcover, Feb 08 2016 *)
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PROG
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(PARI) a(n) = sum(k=1, n-1, k*sigma(k)*sigma(n-k)); \\ Michel Marcus, Feb 02 2014
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CROSSREFS
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Cf. A000385, A000477, A000499, A259692, A259693, A259694, A259695, A259696.
Cf. A000203 (sigma_1), A001158 (sigma_3), A064987.
Cf. A143128, A340793.
Sequence in context: A326278 A014816 A147691 * A067989 A002881 A268803
Adjacent sequences: A000438 A000439 A000440 * A000442 A000443 A000444
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Sean A. Irvine, Nov 14 2010
a(1)=0 prepended by Michel Marcus, Feb 02 2014
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STATUS
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approved
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