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A321812 Sum of 8th powers of odd divisors of n. 4
1, 1, 6562, 1, 390626, 6562, 5764802, 1, 43053283, 390626, 214358882, 6562, 815730722, 5764802, 2563287812, 1, 6975757442, 43053283, 16983563042, 390626, 37828630724, 214358882, 78310985282, 6562, 152588281251, 815730722, 282472589764 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
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) = A013956(A000265(n)) = sigma_8(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)^8*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^(8*e+8)-1)/(p^8-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^9, where c = zeta(9)/18 = 0.0556671... . (End)
MATHEMATICA
a[n_] := DivisorSum[n, #^8 &, OddQ[#] &]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)
PROG
(PARI) apply( A321812(n)=sigma(n>>valuation(n, 2), 8), [1..30]) \\ M. F. Hasler, Nov 26 2018
(Python)
from sympy import divisor_sigma
def A321812(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(), 8)) # Chai Wah Wu, Jul 16 2022
CROSSREFS
Column k=8 of A285425.
Cf. A050999, A051000, A051001, A051002, A321810 - A321816 (analog for 2nd .. 12th powers).
Cf. A321543 - A321565, A321807 - A321836 for related sequences.
Sequence in context: A017432 A017564 A352036 * A031579 A288884 A031759
KEYWORD
nonn,mult
AUTHOR
N. J. A. Sloane, Nov 24 2018
STATUS
approved

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Last modified March 28 16:34 EDT 2024. Contains 371254 sequences. (Running on oeis4.)