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A000477 a(n) = Sum_{k=1..n-1} k^2*sigma(k)*sigma(n-k).
(Formerly M4973 N2135)
2
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 (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^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

EXAMPLE

G.f. = x^2 + 15*x^3 + 76*x^4 + 275*x^5 + 720*x^6 + 1666*x^7 + 3440*x^8 + ...

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-Fran├žois Alcover, Feb 08 2016 *)

PROG

(PARI) a(n) = sum(k=1, n-1, k^2*sigma(k)*sigma(n-k)); \\ Michel Marcus, Feb 02 2014

CROSSREFS

Cf. A000441, A000499.

Sequence in context: A247264 A212093 A212241 * A302376 A041430 A156941

Adjacent sequences:  A000474 A000475 A000476 * A000478 A000479 A000480

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Sean A. Irvine, Nov 14 2010

a(1)=0 prepended by Michel Marcus, Feb 02 2014

STATUS

approved

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Last modified March 20 15:43 EDT 2019. Contains 321345 sequences. (Running on oeis4.)