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A352033
Sum of the 5th powers of the odd proper divisors of n.
11
0, 1, 1, 1, 1, 244, 1, 1, 244, 3126, 1, 244, 1, 16808, 3369, 1, 1, 59293, 1, 3126, 17051, 161052, 1, 244, 3126, 371294, 59293, 16808, 1, 762744, 1, 1, 161295, 1419858, 19933, 59293, 1, 2476100, 371537, 3126, 1, 4101152, 1, 161052, 821793, 6436344, 1, 244, 16808, 9768751
OFFSET
1,6
FORMULA
a(n) = Sum_{d|n, d<n, d odd} d^5.
G.f.: Sum_{k>=1} (2*k-1)^5 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022
From Amiram Eldar, Oct 11 2023: (Start)
a(n) = A051002(n) - n^5*A000035(n).
Sum_{k=1..n} a(k) ~ c * n^6, where c = (zeta(6)-1)/12 = 0.0014452551... . (End)
EXAMPLE
a(10) = 3126; a(10) = Sum_{d|10, d<10, d odd} d^5 = 1^5 + 5^5 = 3126.
MATHEMATICA
Table[Total[Select[Most[Divisors[n]], OddQ]^5], {n, 50}] (* Harvey P. Dale, May 01 2023 *)
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 *)
CROSSREFS
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).
Sequence in context: A017561 A164657 A151638 * A248137 A243774 A051002
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Mar 01 2022
STATUS
approved