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

1,2

Links

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

Formula

a(n) = n^7 * Sum_{d|n, d<n, d odd} 1 / d^7.

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)

Example

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

Mathematica

A352053[n_]:=DivisorSum[n, 1/#^7&, #<n&&OddQ[#]&]n^7; Array[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 *)

Prog

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

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.

Sequence in context: A017678 A123253 A001015 * A050754 A351605 A343287

Adjacent sequences: A352050 A352051 A352052 * A352054 A352055 A352056

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Mar 01 2022

Status

approved