A352032 Sum of the 4th powers of the odd proper divisors of n.

0, 1, 1, 1, 1, 82, 1, 1, 82, 626, 1, 82, 1, 2402, 707, 1, 1, 6643, 1, 626, 2483, 14642, 1, 82, 626, 28562, 6643, 2402, 1, 51332, 1, 1, 14723, 83522, 3027, 6643, 1, 130322, 28643, 626, 1, 196964, 1, 14642, 57893, 279842, 1, 82, 2402, 391251, 83603, 28562, 1, 538084, 15267

Offset: 1

Author

Wesley Ivan Hurt, Mar 01 2022

Keywords

• nonn
• easy

Examples

			a(10) = 626; a(10) = Sum_{d|10, d<10, d odd} d^4 = 1^4 + 5^4 = 626.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to sums of divisors.

Crossrefs

Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), A091570 (k=1), A351647 (k=2), A352031 (k=3), this sequence (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).

Cf. A000035, A013663, A051001.

Programs

Mathematica

f[2, e_] := 1; f[p_, e_] := (p^(4*e+4) - 1)/(p^4 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^4, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)

Formula

a(n) = Sum_{d|n, d

G.f.: Sum_{k>=1} (2*k-1)^4 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022

From Amiram Eldar, Oct 11 2023: (Start)

a(n) = A051001(n) - n^4*A000035(n).

Sum_{k=1..n} a(k) ~ c * n^5, where c = (zeta(5)-1)/10 = 0.0036927755... . (End)

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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