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A321816 Sum of 12th powers of odd divisors of n.

%I A321816 #28 Feb 16 2025 08:33:57
%S A321816 1,1,531442,1,244140626,531442,13841287202,1,282430067923,244140626,
%T A321816 3138428376722,531442,23298085122482,13841287202,129746582562692,1,
%U A321816 582622237229762,282430067923,2213314919066162,244140626,7355841353205284,3138428376722
%N A321816 Sum of 12th powers of odd divisors of n.
%H A321816 Seiichi Manyama, <a href="/A321816/b321816.txt">Table of n, a(n) for n = 1..10000</a>
%H A321816 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 A321816 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OddDivisorFunction.html">Odd Divisor Function</a>.
%H A321816 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321816 a(n) = A013960(A000265(n)) = sigma_12(odd part of n); in particular, a(2^k) = 1 for all k >= 0. - _M. F. Hasler_, Nov 26 2018
%F A321816 G.f.: Sum_{k>=1} (2*k - 1)^12*x^(2*k-1)/(1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Dec 22 2018
%F A321816 From _Amiram Eldar_, Nov 02 2022: (Start)
%F A321816 Multiplicative with a(2^e) = 1 and a(p^e) = (p^(12*e+12)-1)/(p^12-1) for p > 2.
%F A321816 Sum_{k=1..n} a(k) ~ c * n^13, where c = zeta(13)/26 = 0.0384662... . (End)
%t A321816 a[n_] := DivisorSum[n, #^12&, OddQ[#]&]; Array[a, 20] (* _Amiram Eldar_, Dec 07 2018 *)
%o A321816 (PARI) apply( A321816(n)=sigma(n>>valuation(n,2),12), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%o A321816 (Python)
%o A321816 from sympy import divisor_sigma
%o A321816 def A321816(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),12)) # _Chai Wah Wu_, Jul 16 2022
%Y A321816 Column k=12 of A285425.
%Y A321816 Cf. A050999, A051000, A051001, A051002, A321810 - A321815 (analog for 2nd .. 11th powers).
%Y A321816 Cf. A321543 - A321565, A321807 - A321836 for related sequences.
%Y A321816 Cf. A000265, A013671, A013960.
%K A321816 nonn,mult
%O A321816 1,3
%A A321816 _N. J. A. Sloane_, Nov 24 2018