OFFSET
1,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Eric Weisstein's World of Mathematics, Odd Divisor Function.
FORMULA
a(n) = A013957(A000265(n)) = sigma_9(odd part of n); in particular, a(2^k) = 1 for all k >= 0. - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (2*k - 1)^9*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 07 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 1 and a(p^e) = (p^(9*e+9)-1)/(p^9-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^10, where c = zeta(10)/20 = Pi^10/1871100 = 0.0500497... . (End)
MATHEMATICA
a[n_] := DivisorSum[n, #^9 &, OddQ[#] &]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)
PROG
(PARI) apply( A321813(n)=sigma(n>>valuation(n, 2), 9), [1..30]) \\ M. F. Hasler, Nov 26 2018
(Python)
from sympy import divisor_sigma
def A321813(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(), 9)) # Chai Wah Wu, Jul 16 2022
CROSSREFS
Column k=9 of A285425.
KEYWORD
nonn,mult
AUTHOR
N. J. A. Sloane, Nov 24 2018
STATUS
approved