A351647 Sum of the squares of the odd proper divisors of n.

0, 1, 1, 1, 1, 10, 1, 1, 10, 26, 1, 10, 1, 50, 35, 1, 1, 91, 1, 26, 59, 122, 1, 10, 26, 170, 91, 50, 1, 260, 1, 1, 131, 290, 75, 91, 1, 362, 179, 26, 1, 500, 1, 122, 341, 530, 1, 10, 50, 651, 299, 170, 1, 820, 147, 50, 371, 842, 1, 260, 1, 962, 581, 1, 195, 1220, 1, 290

Offset

1,6

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Index entries for sequences related to sums of divisors.

Formula

a(n) = Sum_{d|n, d<n, d odd} d^2.

G.f.: Sum_{k>=1} (2*k-1)^2 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022

From Amiram Eldar, Oct 11 2023: (Start)

a(n) = A050999(n) - n^2*A000035(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(3)-1)/6 = 0.0336761505... . (End)

Example

a(10) = 26; a(10) = Sum_{d|10, d<10, d odd} d^2 = 1^2 + 5^2 = 26.

Mathematica

f[2, e_] := 1; f[p_, e_] := (p^(2*e+2) - 1)/(p^2 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^2, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)

Prog

(PARI) a(n) = sumdiv(n, d, if ((d%2) && (d<n), d^2)); \\ Michel Marcus, Mar 02 2022

Crossrefs

Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), A091570 (k=1), this sequence (k=2), A352031 (k=3), A352032 (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).

Cf. A000035, A002117, A050999.

Sequence in context: A010176 A347161 A347173 * A168644 A168620 A143683

Adjacent sequences: A351644 A351645 A351646 * A351648 A351649 A351650

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Mar 01 2022

Status

approved