|
%I #20 Oct 11 2023 03:54:04
%S 0,1,1,1,1,244,1,1,244,3126,1,244,1,16808,3369,1,1,59293,1,3126,17051,
%T 161052,1,244,3126,371294,59293,16808,1,762744,1,1,161295,1419858,
%U 19933,59293,1,2476100,371537,3126,1,4101152,1,161052,821793,6436344,1,244,16808,9768751
%N Sum of the 5th powers of the odd proper divisors of n.
%H Seiichi Manyama, <a href="/A352033/b352033.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.
%F a(n) = Sum_{d|n, d<n, d odd} d^5.
%F G.f.: Sum_{k>=1} (2*k-1)^5 * x^(4*k-2) / (1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Mar 02 2022
%F From _Amiram Eldar_, Oct 11 2023: (Start)
%F a(n) = A051002(n) - n^5*A000035(n).
%F Sum_{k=1..n} a(k) ~ c * n^6, where c = (zeta(6)-1)/12 = 0.0014452551... . (End)
%e a(10) = 3126; a(10) = Sum_{d|10, d<10, d odd} d^5 = 1^5 + 5^5 = 3126.
%t Table[Total[Select[Most[Divisors[n]],OddQ]^5],{n,50}] (* _Harvey P. Dale_, May 01 2023 *)
%t f[2, e_] := 1; f[p_, e_] := (p^(5*e+5) - 1)/(p^5 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^5, 0]; Array[a, 60] (* _Amiram Eldar_, Oct 11 2023 *)
%Y 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), this sequence (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).
%Y Cf. A000035, A013664, A051002.
%K nonn,easy
%O 1,6
%A _Wesley Ivan Hurt_, Mar 01 2022
|
