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A352034 Sum of the 6th powers of the odd proper divisors of n. 11
0, 1, 1, 1, 1, 730, 1, 1, 730, 15626, 1, 730, 1, 117650, 16355, 1, 1, 532171, 1, 15626, 118379, 1771562, 1, 730, 15626, 4826810, 532171, 117650, 1, 11406980, 1, 1, 1772291, 24137570, 133275, 532171, 1, 47045882, 4827539, 15626, 1, 85884500, 1, 1771562, 11938421, 148035890 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
LINKS
FORMULA
a(n) = Sum_{d|n, d<n, d odd} d^6.
G.f.: Sum_{k>=1} (2*k-1)^6 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022
For odd n >1, a(n) = A321810(n)-n^6; for even n, a(n) = A321810(n). - R. J. Mathar, Aug 15 2023
Sum_{k=1..n} a(k) ~ c * n^7, where c = (zeta(7)-1)/14 = 0.0005963769... . - Amiram Eldar, Oct 11 2023
EXAMPLE
a(10) = 15626; a(10) = Sum_{d|10, d<10, d odd} d^6 = 1^6 + 5^6 = 15626.
MATHEMATICA
f[2, e_] := 1; f[p_, e_] := (p^(6*e+6) - 1)/(p^6 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^6, 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), A352033 (k=5), this sequence (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).
Sequence in context: A277185 A343693 A357960 * A321810 A252526 A044880
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Mar 01 2022
STATUS
approved

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Last modified June 12 10:35 EDT 2024. Contains 373331 sequences. (Running on oeis4.)