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A352038 Sum of the 10th powers of the odd proper divisors of n.

%I A352038 #23 Oct 11 2023 03:54:43
%S A352038 0,1,1,1,1,59050,1,1,59050,9765626,1,59050,1,282475250,9824675,1,1,
%T A352038 3486843451,1,9765626,282534299,25937424602,1,59050,9765626,
%U A352038 137858491850,3486843451,282475250,1,576660215300,1,1,25937483651,2015993900450,292240875,3486843451
%N A352038 Sum of the 10th powers of the odd proper divisors of n.
%H A352038 Seiichi Manyama, <a href="/A352038/b352038.txt">Table of n, a(n) for n = 1..10000</a>
%H A352038 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.
%F A352038 a(n) = Sum_{d|n, d<n, d odd} d^10.
%F A352038 G.f.: Sum_{k>=1} (2*k-1)^10 * x^(4*k-2) / (1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Mar 02 2022
%F A352038 From _Amiram Eldar_, Oct 11 2023: (Start)
%F A352038 a(n) = A321814(n) - n^10*A000035(n).
%F A352038 Sum_{k=1..n} a(k) ~ c * n^11, where c = (zeta(11)-1)/22 = 0.0000224631... . (End)
%e A352038 a(10) = 9765626; a(10) = Sum_{d|10, d<10, d odd} d^10 = 1^10 + 5^10 = 9765626.
%t A352038 f[2, e_] := 1; f[p_, e_] := (p^(10*e+10) - 1)/(p^10 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^10, 0]; Array[a, 60] (* _Amiram Eldar_, Oct 11 2023 *)
%o A352038 (Python)
%o A352038 from math import prod
%o A352038 from sympy import factorint
%o A352038 def A352038(n): return prod((p**(10*(e+1))-1)//(p**10-1) for p, e in factorint(n).items() if p > 2) - (n**10 if n % 2 else 0) # _Chai Wah Wu_, Mar 01 2022
%Y A352038 Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), A091570 (k=1), A351647 (k=2), A352031 (k=3), A352032 (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), this sequence (k=10).
%Y A352038 Cf. A000035, A013669, A321814.
%K A352038 nonn,easy
%O A352038 1,6
%A A352038 _Wesley Ivan Hurt_, Mar 01 2022