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A051000
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Sum of cubes of odd divisors of n.
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23
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1, 1, 28, 1, 126, 28, 344, 1, 757, 126, 1332, 28, 2198, 344, 3528, 1, 4914, 757, 6860, 126, 9632, 1332, 12168, 28, 15751, 2198, 20440, 344, 24390, 3528, 29792, 1, 37296, 4914, 43344, 757, 50654, 6860, 61544, 126, 68922, 9632, 79508, 1332, 95382, 12168, 103824, 28
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OFFSET
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1,3
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COMMENTS
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The sum of cubes of even divisors of 2*k equals 8*A001158(k), and the sum of cubes of even divisors of 2*k-1 vanishes, for k >= 1. - Wolfdieter Lang, Jan 07 2017
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LINKS
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FORMULA
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Dirichlet g.f.: (1-2^(3-s))*zeta(s)*zeta(s-3). Dirichlet convolution of (-1)^n*A176415(n) and A000578. - R. J. Mathar, Apr 06 2011
G.f.: Sum_{k>=1} (2*k - 1)^3*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Jan 04 2017
Multiplicative with a(2^e) = 1 and a(p^e) = (p^(3*e+3)-1)/(p^3-1) for p > 2. - Amiram Eldar, Sep 14 2020
G.f.: Sum_{n >= 1} x^n*(1 + 23*x^(2*n) + 23*x^(4*n) + x^(6*n))/(1 - x^(2*n))^4. See row 4 of A060187. - Peter Bala, Dec 20 2021
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MATHEMATICA
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Table[Total[Select[Divisors[n], OddQ]^3], {n, 50}] (* Harvey P. Dale, Jun 28 2012 *)
f[2, e_] := 1; f[p_, e_] := (p^(3*e + 3) - 1)/(p^3 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)
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PROG
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(Haskell)
a051000 = sum . map (^ 3) . a182469_row
(PARI) a(n) = sumdiv(n, d, (d%2)*d^3); \\ Michel Marcus, Jan 04 2017
(Python)
from sympy import divisor_sigma
def A051000(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(), 3)) # Chai Wah Wu, Jul 16 2022
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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