A352054 Sum of the 8th powers of the divisor complements of the odd proper divisors of n.

0, 256, 6561, 65536, 390625, 1679872, 5764801, 16777216, 43053282, 100000256, 214358881, 430047232, 815730721, 1475789312, 2563287811, 4294967296, 6975757441, 11021640448, 16983563041, 25600065536, 37828630723, 54875873792, 78310985281, 110092091392, 152588281250

Offset

1,2

Links

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

Formula

a(n) = n^8 * Sum_{d|n, d<n, d odd} 1 / d^8.

G.f.: Sum_{k>=2} k^8 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 19 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A321812(n) * A006519(n)^8 - A000035(n).

Sum_{k=1..n} a(k) = c * n^9 / 9, where c = 511*zeta(9)/512 = 1.0000513451... . (End)

Example

a(10) = 10^8 * Sum_{d|10, d<10, d odd} 1 / d^8 = 10^8 * (1/1^8 + 1/5^8) = 100000256.

Mathematica

A352054[n_]:=DivisorSum[n, 1/#^8&, #<n&&OddQ[#]&]n^8; Array[A352054, 50] (* Paolo Xausa, Aug 09 2023 *)
a[n_] := DivisorSigma[-8, n/2^IntegerExponent[n, 2]] * n^8 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

Prog

(PARI) a(n) = n^8 * sigma(n >> valuation(n, 2), -8) - 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), A352053 (k=7), this sequence (k=8), A352055 (k=9), A352056 (k=10).

Cf. A000035, A006519, A013667, A321812.

Sequence in context: A017680 A210840 A001016 * A351606 A343288 A050755

Adjacent sequences: A352051 A352052 A352053 * A352055 A352056 A352057

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Mar 01 2022

Status

approved