A000477 a(n) = Sum_{k=1..n-1} k^2*sigma(k)*sigma(n-k).
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
Offset: 1
Keywords
Examples
G.f. = x^2 + 15*x^3 + 76*x^4 + 275*x^5 + 720*x^6 + 1666*x^7 + 3440*x^8 + ...
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).
- Jacques 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
- Jacques Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]
Crossrefs
Programs
-
Maple
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
-
Mathematica
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 *) -
PARI
a(n) = sum(k=1, n-1, k^2*sigma(k)*sigma(n-k)); \\ Michel Marcus, Feb 02 2014
-
PARI
a(n) = my(f = factor(n)); ((n^2 - 4*n^3) * sigma(f) + 3*n^2 * sigma(f, 3)) / 24; \\ Amiram Eldar, Jan 04 2025
Formula
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
Extensions
More terms from Sean A. Irvine, Nov 14 2010
a(1)=0 prepended by Michel Marcus, Feb 02 2014