A352053 Sum of the 7th powers of the divisor complements of the odd proper divisors of n.

0, 128, 2187, 16384, 78125, 280064, 823543, 2097152, 4785156, 10000128, 19487171, 35848192, 62748517, 105413632, 170939687, 268435456, 410338673, 612500096, 893871739, 1280016384, 1801914271, 2494358016, 3404825447, 4588568576, 6103593750, 8031810304, 10465138359

Offset: 1

Author

Wesley Ivan Hurt, Mar 01 2022

Keywords

• nonn
• easy

Examples

			a(10) = 10^7 * Sum_{d|10, d<10, d odd} 1/d^7 = 10^7 * (1/1^7 + 1/5^7) = 10000128.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000

Crossrefs

Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), A352047 (k=1), A352048 (k=2), A352049 (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), this sequence (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Cf. A000035, A006519, A013666, A321811.

Programs

Mathematica

A352053[n_]:=DivisorSum[n,1/#^7&,#A352053,50] (* Paolo Xausa, Aug 09 2023 *)
a[n_] := DivisorSigma[-7, n/2^IntegerExponent[n, 2]] * n^7 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

PARI

a(n) = n^7 * sigma(n >> valuation(n, 2), -7) - n % 2; \\ Amiram Eldar, Oct 13 2023

Formula

a(n) = n^7 * Sum_{d|n, d

G.f.: Sum_{k>=2} k^7 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 18 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A321811(n) * A006519(n)^7 - A000035(n).

Sum_{k=1..n} a(k) = c * n^8 / 8, where c = 255*zeta(8)/256 = 1.000155179... . (End)

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

Content is available under The OEIS End-User License Agreement.