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A051000 Sum of cubes of odd divisors of n.

%I A051000 #60 Feb 16 2025 08:32:41
%S A051000 1,1,28,1,126,28,344,1,757,126,1332,28,2198,344,3528,1,4914,757,6860,
%T A051000 126,9632,1332,12168,28,15751,2198,20440,344,24390,3528,29792,1,37296,
%U A051000 4914,43344,757,50654,6860,61544,126,68922,9632,79508,1332,95382,12168,103824,28
%N A051000 Sum of cubes of odd divisors of n.
%C A051000 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
%C A051000 Sum_{k>=1} 1/a(k) diverges. - _Vaclav Kotesovec_, Sep 21 2020
%H A051000 Reinhard Zumkeller, <a href="/A051000/b051000.txt">Table of n, a(n) for n = 1..10000</a>
%H A051000 John A. Ewell, <a href="http://www.pphmj.com/abstract/2504.htm">On a relation between two divisor functions</a>, JP Journal of Algebra, Number Theory and Applications, Vol. 7, No. 2 (2007), pp. 241-243.
%H A051000 J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&amp;pg=RA1-PA1">On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares</a>, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
%H A051000 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OddDivisorFunction.html">Odd Divisor Function</a>.
%H A051000 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A051000 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
%F A051000 a(n) = Sum_{k=1..A001227(n)} A182469(n,k)^3. - _Reinhard Zumkeller_, May 01 2012
%F A051000 G.f.: Sum_{k>=1} (2*k - 1)^3*x^(2*k-1)/(1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Jan 04 2017
%F A051000 Sum_{k=1..n} a(k) ~ Pi^4 * n^4 / 720. - _Vaclav Kotesovec_, Jan 31 2019
%F A051000 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
%F A051000 For k>=0, a(2^k) = 1. - _Vaclav Kotesovec_, Sep 21 2020
%F A051000 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
%F A051000 a(n) = Sum_{k=0..n-1} A000203(2*n-2*k-1)*A000203(2*k+1)/A006519(n)^3 (Ewell, 2007). - _Amiram Eldar_, Feb 24 2024
%t A051000 Table[Total[Select[Divisors[n],OddQ]^3],{n,50}] (* _Harvey P. Dale_, Jun 28 2012 *)
%t A051000 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 *)
%o A051000 (Haskell)
%o A051000 a051000 = sum . map (^ 3) . a182469_row
%o A051000 -- _Reinhard Zumkeller_, May 01 2012
%o A051000 (PARI) a(n) = sumdiv(n, d, (d%2)*d^3); \\ _Michel Marcus_, Jan 04 2017
%o A051000 (Python)
%o A051000 from sympy import divisor_sigma
%o A051000 def A051000(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),3)) # _Chai Wah Wu_, Jul 16 2022
%Y A051000 Cf. A000203, A000593, A001227, A001158, A006519, A050999, A051001, A051002.
%K A051000 nonn,mult
%O A051000 1,3
%A A051000 _Eric W. Weisstein_