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A352054
Sum of the 8th powers of the divisor complements of the odd proper divisors of n.
11
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
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).
Sequence in context: A017680 A210840 A001016 * A351606 A343288 A050755
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Mar 01 2022
STATUS
approved