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 A000499 a(n) = Sum_{k=1..n-1} k^3*sigma(k)*sigma(n-k). (Formerly M5193 N2257) 8
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 REFERENCES 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. LINKS 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] FORMULA 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 EXAMPLE G.f. = x^2 + 27*x^3 + 184*x^4 + 875*x^5 + 2700*x^6 + 7546*x^7 + 17600*x^8 + ... MAPLE 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); MATHEMATICA 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 *) PROG (PARI) a(n) = sum(k=1, n-1, k^3*sigma(k)*sigma(n-k)); \\ Michel Marcus, Feb 02 2014 CROSSREFS 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 KEYWORD nonn AUTHOR EXTENSIONS More terms and 0 prepended by Michel Marcus, Feb 02 2014 STATUS approved

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Last modified March 31 14:55 EDT 2023. Contains 361666 sequences. (Running on oeis4.)