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A352056
Sum of the 10th powers of the divisor complements of the odd proper divisors of n.
11
0, 1024, 59049, 1048576, 9765625, 60467200, 282475249, 1073741824, 3486843450, 10000001024, 25937424601, 61918412800, 137858491849, 289254656000, 576660215299, 1099511627776, 2015993900449, 3570527693824, 6131066257801, 10240001048576, 16680163512499
OFFSET
1,2
FORMULA
a(n) = n^10 * Sum_{d|n, d<n, d odd} 1 / d^10.
G.f.: Sum_{k>=2} k^10 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 19 2023
From Amiram Eldar, Oct 13 2023: (Start)
a(n) = A321814(n) * A006519(n)^10 - A000035(n).
Sum_{k=1..n} a(k) = c * n^11 / 11, where c = 2047*zeta(11)/2048 = 1.00000566605... . (End)
EXAMPLE
a(10) = 10^10 * Sum_{d|10, d<10, d odd} 1 / d^10 = 10^10 * (1/1^10 + 1/5^10) = 10000001024.
MATHEMATICA
A352056[n_]:=DivisorSum[n, 1/#^10&, #<n&&OddQ[#]&]n^10; Array[A352056, 50] (* Paolo Xausa, Aug 10 2023 *)
a[n_] := DivisorSigma[-10, n/2^IntegerExponent[n, 2]] * n^10 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)
PROG
(PARI) a(n) = n^10 * sigma(n >> valuation(n, 2), -10) - 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), A352055 (k=9), this sequence (k=10).
Sequence in context: A016901 A017684 A008454 * A351608 A030629 A056587
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Mar 01 2022
STATUS
approved