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A051001 Sum of 4th powers of odd divisors of n.

%I A051001 #41 Feb 16 2025 08:32:41
%S A051001 1,1,82,1,626,82,2402,1,6643,626,14642,82,28562,2402,51332,1,83522,
%T A051001 6643,130322,626,196964,14642,279842,82,391251,28562,538084,2402,
%U A051001 707282,51332,923522,1,1200644,83522,1503652,6643,1874162,130322,2342084,626,2825762,196964
%N A051001 Sum of 4th powers of odd divisors of n.
%H A051001 Robert Israel, <a href="/A051001/b051001.txt">Table of n, a(n) for n = 1..10000</a>
%H A051001 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 A051001 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OddDivisorFunction.html">Odd Divisor Function</a>.
%H A051001 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>
%F A051001 Dirichlet g.f. (1-2^(4-s))*zeta(s)*zeta(s-4). - _R. J. Mathar_, Apr 06 2011
%F A051001 G.f.: Sum_{k>=1} (2*k - 1)^4*x^(2*k-1)/(1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Jan 04 2017
%F A051001 a(n) = A001159(A000265(n)). - _Robert Israel_, Jan 05 2017
%F A051001 Multiplicative with a(2^e) = 1 and a(p^e) = (p^(4*e+4)-1)/(p^4-1) for p > 2. - _Amiram Eldar_, Sep 14 2020
%F A051001 Sum_{k=1..n} a(k) ~ zeta(5)*n^5/10. - _Vaclav Kotesovec_, Sep 24 2020
%F A051001 G.f.: Sum_{n >= 1} x^n*(1 + 76*x^(2*n) + 230*x^(4*n) + 76*x^(6*n) + x^(8*n))/(1 - x^(2*n))^5. See row 5 of A060187. - _Peter Bala_, Dec 20 2021
%p A051001 f:= proc(n) add(x^4, x = numtheory:-divisors(n/2^padic:-ordp(n,2))) end proc:
%p A051001 map(f, [$1..100]); # _Robert Israel_, Jan 05 2017
%t A051001 Table[Total[Select[Divisors[n],OddQ]^4],{n,40}] (* _Harvey P. Dale_, Oct 02 2014 *)
%t A051001 f[2, e_] := 1; f[p_, e_] := (p^(4*e + 4) - 1)/(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 14 2020 *)
%o A051001 (PARI) a(n) = sumdiv(n , d, (d%2)*d^4); \\ _Michel Marcus_, Jan 14 2014
%o A051001 (Python)
%o A051001 from sympy import divisor_sigma
%o A051001 def A051001(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),4)) # _Chai Wah Wu_, Jul 16 2022
%Y A051001 Cf. A000265, A000593, A001227, A001159, A050999, A051000, A051002.
%K A051001 nonn,mult,look
%O A051001 1,3
%A A051001 _Eric W. Weisstein_