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A000499
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a(n) = Sum_{k=1..n-1} k^3*sigma(k)*sigma(n-k).
(Formerly M5193 N2257)
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8
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0, 1, 27, 184, 875, 2700, 7546, 17600, 35721, 72750, 126445, 223776, 353717, 595448, 843750, 1349120, 1827636, 2808837, 3600975, 5306000, 6667920, 9599172, 11509982, 16416000, 19015625, 26605670, 30902310, 41686848, 46948825, 64233000, 70306760, 94089216
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OFFSET
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1,3
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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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John Cerkan, Table of n, a(n) for n = 1..10000
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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a(n) = Sum_{k=1..n-1} k^3*sigma(k)*sigma(n-k). - Michel Marcus, Feb 02 2014
a(n) = (n^3/24 - n^4/8)*sigma_1(n) + (n^3/12)*sigma_3(n). - Ridouane Oudra, Sep 15 2020
Sum_{k=1..n} a(k) ~ Pi^4 * n^7 / 7560. - Vaclav Kotesovec, Aug 08 2022
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EXAMPLE
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G.f. = x^2 + 27*x^3 + 184*x^4 + 875*x^5 + 2700*x^6 + 7546*x^7 + 17600*x^8 + ...
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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(3);
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MATHEMATICA
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a[n_] := Sum[k^3*DivisorSigma[1, k]*DivisorSigma[1, n - k], {k, 1, n - 1}]; Array[a, 32] (* Jean-François Alcover, Feb 09 2016 *)
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PROG
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(PARI) a(n) = sum(k=1, n-1, k^3*sigma(k)*sigma(n-k)); \\ Michel Marcus, Feb 02 2014
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CROSSREFS
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Cf. A000385, A000441, A000477, A259692, A259693, A259694, A259695, A259696.
Cf. A000203 (sigma_1), A001158 (sigma_3).
Sequence in context: A224454 A258637 A228463 * A042416 A216108 A216110
Adjacent sequences: A000496 A000497 A000498 * A000500 A000501 A000502
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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 and 0 prepended by Michel Marcus, Feb 02 2014
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STATUS
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approved
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