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A321814 Sum of 10th powers of odd divisors of n.

%I A321814 #25 Feb 16 2025 08:33:57
%S A321814 1,1,59050,1,9765626,59050,282475250,1,3486843451,9765626,25937424602,
%T A321814 59050,137858491850,282475250,576660215300,1,2015993900450,3486843451,
%U A321814 6131066257802,9765626,16680163512500,25937424602,41426511213650,59050
%N A321814 Sum of 10th powers of odd divisors of n.
%H A321814 Seiichi Manyama, <a href="/A321814/b321814.txt">Table of n, a(n) for n = 1..10000</a>
%H A321814 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 A321814 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OddDivisorFunction.html">Odd Divisor Function</a>.
%H A321814 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321814 a(n) = A013958(A000265(n)) = sigma_10(odd part of n); in particular, a(2^k) = 1 for all k >= 0. - _M. F. Hasler_, Nov 26 2018
%F A321814 G.f.: Sum_{k>=1} (2*k - 1)^10*x^(2*k-1)/(1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Dec 07 2018
%F A321814 From _Amiram Eldar_, Nov 02 2022: (Start)
%F A321814 Multiplicative with a(2^e) = 1 and a(p^e) = (p^(10*e+10)-1)/(p^10-1) for p > 2.
%F A321814 Sum_{k=1..n} a(k) ~ c * n^11, where c = zeta(11)/22 = 0.045477... . (End)
%t A321814 a[n_] := DivisorSum[n, #^10 &, OddQ[#] &]; Array[a, 20] (* _Amiram Eldar_, Dec 07 2018 *)
%o A321814 (PARI) apply( A321814(n)=sigma(n>>valuation(n,2),10), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%o A321814 (Python)
%o A321814 from sympy import divisor_sigma
%o A321814 def A321814(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),10)) # _Chai Wah Wu_, Jul 16 2022
%Y A321814 Column k=10 of A285425.
%Y A321814 Cf. A050999, A051000, A051001, A051002, A321810 - A321816 (analog for 2nd .. 12th powers).
%Y A321814 Cf. A321543 - A321565, A321807 - A321836 for related sequences.
%Y A321814 Cf. A000265, A013669, A013958.
%K A321814 nonn,mult
%O A321814 1,3
%A A321814 _N. J. A. Sloane_, Nov 24 2018