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A351647
Sum of the squares of the odd proper divisors of n.
12
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
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).
Sequence in context: A010176 A347161 A347173 * A168644 A168620 A143683
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Mar 01 2022
STATUS
approved