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A352053 Sum of the 7th powers of the divisor complements of the odd proper divisors of n.

%I A352053 #18 Oct 13 2023 06:52:08
%S A352053 0,128,2187,16384,78125,280064,823543,2097152,4785156,10000128,
%T A352053 19487171,35848192,62748517,105413632,170939687,268435456,410338673,
%U A352053 612500096,893871739,1280016384,1801914271,2494358016,3404825447,4588568576,6103593750,8031810304,10465138359
%N A352053 Sum of the 7th powers of the divisor complements of the odd proper divisors of n.
%H A352053 Paolo Xausa, <a href="/A352053/b352053.txt">Table of n, a(n) for n = 1..10000</a>
%F A352053 a(n) = n^7 * Sum_{d|n, d<n, d odd} 1 / d^7.
%F A352053 G.f.: Sum_{k>=2} k^7 * x^k / (1 - x^(2*k)). - _Ilya Gutkovskiy_, May 18 2023
%F A352053 From _Amiram Eldar_, Oct 13 2023: (Start)
%F A352053 a(n) = A321811(n) * A006519(n)^7 - A000035(n).
%F A352053 Sum_{k=1..n} a(k) = c * n^8 / 8, where c = 255*zeta(8)/256 = 1.000155179... . (End)
%e A352053 a(10) = 10^7 * Sum_{d|10, d<10, d odd} 1/d^7 = 10^7 * (1/1^7 + 1/5^7) = 10000128.
%t A352053 A352053[n_]:=DivisorSum[n,1/#^7&,#<n&&OddQ[#]&]n^7;Array[A352053,50] (* _Paolo Xausa_, Aug 09 2023 *)
%t A352053 a[n_] := DivisorSigma[-7, n/2^IntegerExponent[n, 2]] * n^7 - Mod[n, 2]; Array[a, 100] (* _Amiram Eldar_, Oct 13 2023 *)
%o A352053 (PARI) a(n) = n^7 * sigma(n >> valuation(n, 2), -7) - n % 2; \\ _Amiram Eldar_, Oct 13 2023
%Y A352053 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), this sequence (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).
%Y A352053 Cf. A000035, A006519, A013666, A321811.
%K A352053 nonn,easy
%O A352053 1,2
%A A352053 _Wesley Ivan Hurt_, Mar 01 2022