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

1,6

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Index entries for sequences related to sums of divisors.

Formula

a(n) = Sum_{d|n, d<n, d odd} d^4.

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)

Example

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

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 *)

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.

Sequence in context: A340398 A347172 A347175 * A051001 A363991 A050678

Adjacent sequences: A352029 A352030 A352031 * A352033 A352034 A352035

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Mar 01 2022

Status

approved