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A276634 Sum of cubes of proper divisors of n. 6
0, 1, 1, 9, 1, 36, 1, 73, 28, 134, 1, 316, 1, 352, 153, 585, 1, 981, 1, 1198, 371, 1340, 1, 2556, 126, 2206, 757, 3160, 1, 4752, 1, 4681, 1359, 4922, 469, 8605, 1, 6868, 2225, 9710, 1, 12600, 1, 12052, 4257, 12176, 1, 20476, 344, 16759, 4941, 19846, 1, 26496, 1457, 25624, 6887, 24398, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

More generally, the Dirichlet generating function for the sum of k-th powers of proper divisors of n is zeta(s-k)*(zeta(s) - 1).

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Proper divisors.

FORMULA

a(n) = 1 if n is prime.

a(p^k) = (p^(3*k) - 1)/(p^3 - 1) for p prime.

Dirichlet g.f.: zeta(s-3)*(zeta(s) - 1).

a(n) = A001158(n) - A000578(n).

A000035(a(n)) = A053867(n).

Sum_{n=1..k} a(n) ~ k^2*(Pi^4*k^2/90 - (k + 1)^2)/4.

G.f.: -x*(1 + 4*x + x^2)/(1 - x)^4 + Sum_{k>=1} k^3*x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 17 2017

EXAMPLE

a(10) = 1^3 + 2^3 + 5^3 = 134, because 10 has 3 proper divisors {1,2,5}.

a(11) = 1^3 = 1, because 11 has 1 proper divisor {1}.

MATHEMATICA

Table[DivisorSigma[3, n] - n^3, {n, 70}]

PROG

(PARI) a(n) = sigma(n, 3) - n^3; \\ Michel Marcus, Sep 08 2016

(MAGMA) [DivisorSigma(3, n) - n^3: n in [1..70]]; // Vincenzo Librandi, Sep 09 2016

CROSSREFS

Cf. A000035, A000578, A001065, A001158, A032741, A053867, A067558.

Sequence in context: A195278 A092477 A019433 * A050312 A295575 A107892

Adjacent sequences:  A276631 A276632 A276633 * A276635 A276636 A276637

KEYWORD

nonn,easy

AUTHOR

Ilya Gutkovskiy, Sep 08 2016

STATUS

approved

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Last modified June 30 06:00 EDT 2022. Contains 354914 sequences. (Running on oeis4.)