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A352056 Sum of the 10th powers of the divisor complements of the odd proper divisors of n.

%I A352056 #18 Oct 13 2023 06:51:47
%S A352056 0,1024,59049,1048576,9765625,60467200,282475249,1073741824,
%T A352056 3486843450,10000001024,25937424601,61918412800,137858491849,
%U A352056 289254656000,576660215299,1099511627776,2015993900449,3570527693824,6131066257801,10240001048576,16680163512499
%N A352056 Sum of the 10th powers of the divisor complements of the odd proper divisors of n.
%H A352056 Paolo Xausa, <a href="/A352056/b352056.txt">Table of n, a(n) for n = 1..10000</a>
%F A352056 a(n) = n^10 * Sum_{d|n, d<n, d odd} 1 / d^10.
%F A352056 G.f.: Sum_{k>=2} k^10 * x^k / (1 - x^(2*k)). - _Ilya Gutkovskiy_, May 19 2023
%F A352056 From _Amiram Eldar_, Oct 13 2023: (Start)
%F A352056 a(n) = A321814(n) * A006519(n)^10 - A000035(n).
%F A352056 Sum_{k=1..n} a(k) = c * n^11 / 11, where c = 2047*zeta(11)/2048 = 1.00000566605... . (End)
%e A352056 a(10) = 10^10 * Sum_{d|10, d<10, d odd} 1 / d^10 = 10^10 * (1/1^10 + 1/5^10) = 10000001024.
%t A352056 A352056[n_]:=DivisorSum[n,1/#^10&,#<n&&OddQ[#]&]n^10;Array[A352056,50] (* _Paolo Xausa_, Aug 10 2023 *)
%t A352056 a[n_] := DivisorSigma[-10, n/2^IntegerExponent[n, 2]] * n^10 - Mod[n, 2]; Array[a, 100] (* _Amiram Eldar_, Oct 13 2023 *)
%o A352056 (PARI) a(n) = n^10 * sigma(n >> valuation(n, 2), -10) - n % 2; \\ _Amiram Eldar_, Oct 13 2023
%Y A352056 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).
%Y A352056 Cf. A000035, A006519, A013669, A321814.
%K A352056 nonn,easy
%O A352056 1,2
%A A352056 _Wesley Ivan Hurt_, Mar 01 2022