A279363 Sum of 4th powers of proper divisors of n.

0, 1, 1, 17, 1, 98, 1, 273, 82, 642, 1, 1650, 1, 2418, 707, 4369, 1, 7955, 1, 10898, 2483, 14658, 1, 26482, 626, 28578, 6643, 41090, 1, 62644, 1, 69905, 14723, 83538, 3027, 133923, 1, 130338, 28643, 174994, 1, 236692, 1, 249170, 57893, 279858, 1, 423794, 2402, 401267, 83603, 485810, 1, 644372, 15267, 659842, 130403, 707298, 1, 1053636

Offset

1,4

Links

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

Eric Weisstein's World of Mathematics, Proper divisors.

Index entries for sequences related to sums of divisors

Formula

a(n) = 1 if n is prime.

a(p^k) = (p^(4*k) - 1)/(p^4 - 1) for p is prime.

Dirichlet g.f.: zeta(s-4)*(zeta(s) - 1).

a(n) = A001159(n) - A000583(n).

G.f.: -x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5 + Sum_{k>=1} k^4 x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 18 2017

Sum_{k=1..n} a(k) ~ (Zeta(5) - 1)*n^5 / 5. - Vaclav Kotesovec, Feb 02 2019

Example

a(10) = 1^4 + 2^4 + 5^4 = 642, because 10 has 3 proper divisors {1,2,5}.
a(11) = 1^4 = 1, because 11 has 1 proper divisor {1}.

Mathematica

Table[DivisorSigma[4, n] - n^4, {n, 60}]

Prog

(PARI) for(n=1, 60, print1(sigma(n, 4) - n^4, ", ")) \\ Indranil Ghosh, Mar 18 2017
(Python)
from sympy.ntheory import divisor_sigma
print([divisor_sigma(n, 4) - n**4 for n in range(1, 61)]) # Indranil Ghosh, Mar 18 2017

Crossrefs

Cf. A000583, A001159.

Cf. A001065, A067558, A276634, A279364.

Sequence in context: A189120 A102292 A264439 * A295576 A376019 A223519

Adjacent sequences: A279360 A279361 A279362 * A279364 A279365 A279366

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Dec 10 2016

Status

approved