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

%I A352054 #17 Oct 13 2023 06:52:01
%S A352054 0,256,6561,65536,390625,1679872,5764801,16777216,43053282,100000256,
%T A352054 214358881,430047232,815730721,1475789312,2563287811,4294967296,
%U A352054 6975757441,11021640448,16983563041,25600065536,37828630723,54875873792,78310985281,110092091392,152588281250
%N A352054 Sum of the 8th powers of the divisor complements of the odd proper divisors of n.
%H A352054 Paolo Xausa, <a href="/A352054/b352054.txt">Table of n, a(n) for n = 1..10000</a>
%F A352054 a(n) = n^8 * Sum_{d|n, d<n, d odd} 1 / d^8.
%F A352054 G.f.: Sum_{k>=2} k^8 * x^k / (1 - x^(2*k)). - _Ilya Gutkovskiy_, May 19 2023
%F A352054 From _Amiram Eldar_, Oct 13 2023: (Start)
%F A352054 a(n) = A321812(n) * A006519(n)^8 - A000035(n).
%F A352054 Sum_{k=1..n} a(k) = c * n^9 / 9, where c = 511*zeta(9)/512 = 1.0000513451... . (End)
%e A352054 a(10) = 10^8 * Sum_{d|10, d<10, d odd} 1 / d^8 = 10^8 * (1/1^8 + 1/5^8) = 100000256.
%t A352054 A352054[n_]:=DivisorSum[n,1/#^8&,#<n&&OddQ[#]&]n^8;Array[A352054,50] (* _Paolo Xausa_, Aug 09 2023 *)
%t A352054 a[n_] := DivisorSigma[-8, n/2^IntegerExponent[n, 2]] * n^8 - Mod[n, 2]; Array[a, 100] (* _Amiram Eldar_, Oct 13 2023 *)
%o A352054 (PARI) a(n) = n^8 * sigma(n >> valuation(n, 2), -8) - n % 2; \\ _Amiram Eldar_, Oct 13 2023
%Y A352054 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).
%Y A352054 Cf. A000035, A006519, A013667, A321812.
%K A352054 nonn,easy
%O A352054 1,2
%A A352054 _Wesley Ivan Hurt_, Mar 01 2022