A352035 - OEIS

%I #19 Oct 11 2023 03:54:54

%S 0,1,1,1,1,2188,1,1,2188,78126,1,2188,1,823544,80313,1,1,4785157,1,

%T 78126,825731,19487172,1,2188,78126,62748518,4785157,823544,1,

%U 170939688,1,1,19489359,410338674,901669,4785157,1,893871740,62750705,78126,1,1801914272,1

%N Sum of the 7th powers of the odd proper divisors of n.

%H Seiichi Manyama, <a href="/A352035/b352035.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^7.

%F G.f.: Sum_{k>=1} (2*k-1)^7 * 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) = A321811(n) - n^7*A000035(n).

%F Sum_{k=1..n} a(k) ~ c * n^8, where c = (zeta(8)-1)/16 = 0.0002548347... . (End)

%e a(10) = 78126; a(10) = Sum_{d|10, d<10, d odd} d^7 = 1^7 + 5^7 = 78126.

%t f[2, e_] := 1; f[p_, e_] := (p^(7*e+7) - 1)/(p^7 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^7, 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), A352033 (k=5), A352034 (k=6), this sequence (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).

%Y Cf. A000035, A013666, A321811.

%K nonn,easy

%O 1,6

%A _Wesley Ivan Hurt_, Mar 01 2022