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A352055
Sum of the 9th powers of the divisor complements of the odd proper divisors of n.
11
0, 512, 19683, 262144, 1953125, 10078208, 40353607, 134217728, 387440172, 1000000512, 2357947691, 5160042496, 10604499373, 20661047296, 38445332183, 68719476736, 118587876497, 198369368576, 322687697779, 512000262144, 794320419871, 1207269218304, 1801152661463, 2641941757952
OFFSET
1,2
FORMULA
a(n) = n^9 * Sum_{d|n, d<n, d odd} 1 / d^9.
G.f.: Sum_{k>=2} k^9 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 19 2023
From Amiram Eldar, Oct 13 2023: (Start)
a(n) = A321813(n) * A006519(n)^9 - A000035(n).
Sum_{k=1..n} a(k) = c * n^10 / 10, where c = 1023*zeta(10)/1024 = 1.0000170413... . (End)
EXAMPLE
a(10) = 10^9 * Sum_{d|10, d<10, d odd} 1 / d^9 = 10^9 * (1/1^9 + 1/5^9) = 1000000512.
MATHEMATICA
A352055[n_]:=DivisorSum[n, 1/#^9&, #<n&&OddQ[#]&]n^9; Array[A352055, 50] (* Paolo Xausa, Aug 10 2023 *)
a[n_] := DivisorSigma[-9, n/2^IntegerExponent[n, 2]] * n^9 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)
PROG
(PARI) a(n) = n^9 * sigma(n >> valuation(n, 2), -9) - 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), A352054 (k=8), this sequence (k=9), A352056 (k=10).
Sequence in context: A351197 A017682 A001017 * A351607 A343289 A050756
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Mar 01 2022
STATUS
approved