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A000477
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a(n) = Sum_{k=1..n-1} k^2*sigma(k)*sigma(n-k).
(Formerly M4973 N2135)
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8
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0, 1, 15, 76, 275, 720, 1666, 3440, 6129, 11250, 17545, 28896, 41405, 65072, 85950, 128960, 162996, 238545, 286995, 404600, 482160, 662112, 756470, 1042560, 1150625, 1549730, 1732590, 2257920, 2443105, 3250800, 3421160, 4452096, 4791600, 6039522, 6296500
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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^2*sigma(k)*sigma(n-k). - Sean A. Irvine, Nov 14 2010
G.f.: x*f(x)*g'(x), where f(x) = Sum_{k>=1} k*x^k/(1 - x^k) and g(x) = Sum_{k>=1} k^2*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, May 02 2018
a(n) = (n^2/24 - n^3/6)*sigma_1(n) + (n^2/8)*sigma_3(n). - Ridouane Oudra, Sep 15 2020
Sum_{k=1..n} a(k) ~ Pi^4 * n^6 / 4320. - Vaclav Kotesovec, May 09 2022
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EXAMPLE
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G.f. = x^2 + 15*x^3 + 76*x^4 + 275*x^5 + 720*x^6 + 1666*x^7 + 3440*x^8 + ...
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MAPLE
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with(numtheory): 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(2); # N. J. A. Sloane, Jul 03 2015
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MATHEMATICA
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a[n_] := Sum[k^2 DivisorSigma[1, k] DivisorSigma[1, n-k], {k, 1, n-1}]; Array[a, 35] (* Jean-François Alcover, Feb 08 2016 *)
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PROG
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(PARI) a(n) = sum(k=1, n-1, k^2*sigma(k)*sigma(n-k)); \\ Michel Marcus, Feb 02 2014
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CROSSREFS
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Cf. A000385, A000441, A000499, A259692, A259693, A259694, A259695, A259696.
Cf. A000203 (sigma_1), A001158 (sigma_3).
Sequence in context: A247264 A212093 A212241 * A302376 A041430 A156941
Adjacent sequences: A000474 A000475 A000476 * A000478 A000479 A000480
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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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