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A352055 Sum of the 9th powers of the divisor complements of the odd proper divisors of n.

%I A352055 #18 Oct 13 2023 06:51:56
%S A352055 0,512,19683,262144,1953125,10078208,40353607,134217728,387440172,
%T A352055 1000000512,2357947691,5160042496,10604499373,20661047296,38445332183,
%U A352055 68719476736,118587876497,198369368576,322687697779,512000262144,794320419871,1207269218304,1801152661463,2641941757952
%N A352055 Sum of the 9th powers of the divisor complements of the odd proper divisors of n.
%H A352055 Paolo Xausa, <a href="/A352055/b352055.txt">Table of n, a(n) for n = 1..10000</a>
%F A352055 a(n) = n^9 * Sum_{d|n, d<n, d odd} 1 / d^9.
%F A352055 G.f.: Sum_{k>=2} k^9 * x^k / (1 - x^(2*k)). - _Ilya Gutkovskiy_, May 19 2023
%F A352055 From _Amiram Eldar_, Oct 13 2023: (Start)
%F A352055 a(n) = A321813(n) * A006519(n)^9 - A000035(n).
%F A352055 Sum_{k=1..n} a(k) = c * n^10 / 10, where c = 1023*zeta(10)/1024 = 1.0000170413... . (End)
%e A352055 a(10) = 10^9 * Sum_{d|10, d<10, d odd} 1 / d^9 = 10^9 * (1/1^9 + 1/5^9) = 1000000512.
%t A352055 A352055[n_]:=DivisorSum[n,1/#^9&,#<n&&OddQ[#]&]n^9;Array[A352055,50] (* _Paolo Xausa_, Aug 10 2023 *)
%t A352055 a[n_] := DivisorSigma[-9, n/2^IntegerExponent[n, 2]] * n^9 - Mod[n, 2]; Array[a, 100] (* _Amiram Eldar_, Oct 13 2023 *)
%o A352055 (PARI) a(n) = n^9 * sigma(n >> valuation(n, 2), -9) - n % 2; \\ _Amiram Eldar_, Oct 13 2023
%Y A352055 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).
%Y A352055 Cf. A000035, A006519, A013668, A321813.
%K A352055 nonn,easy
%O A352055 1,2
%A A352055 _Wesley Ivan Hurt_, Mar 01 2022