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A321813 Sum of 9th powers of odd divisors of n.

%I A321813 #26 Feb 16 2025 08:33:57
%S A321813 1,1,19684,1,1953126,19684,40353608,1,387440173,1953126,2357947692,
%T A321813 19684,10604499374,40353608,38445332184,1,118587876498,387440173,
%U A321813 322687697780,1953126,794320419872,2357947692,1801152661464,19684,3814699218751
%N A321813 Sum of 9th powers of odd divisors of n.
%H A321813 Seiichi Manyama, <a href="/A321813/b321813.txt">Table of n, a(n) for n = 1..10000</a>
%H A321813 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 A321813 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OddDivisorFunction.html">Odd Divisor Function</a>.
%H A321813 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321813 a(n) = A013957(A000265(n)) = sigma_9(odd part of n); in particular, a(2^k) = 1 for all k >= 0. - _M. F. Hasler_, Nov 26 2018
%F A321813 G.f.: Sum_{k>=1} (2*k - 1)^9*x^(2*k-1)/(1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Dec 07 2018
%F A321813 From _Amiram Eldar_, Nov 02 2022: (Start)
%F A321813 Multiplicative with a(2^e) = 1 and a(p^e) = (p^(9*e+9)-1)/(p^9-1) for p > 2.
%F A321813 Sum_{k=1..n} a(k) ~ c * n^10, where c = zeta(10)/20 = Pi^10/1871100 = 0.0500497... . (End)
%t A321813 a[n_] := DivisorSum[n, #^9 &, OddQ[#] &]; Array[a, 20] (* _Amiram Eldar_, Dec 07 2018 *)
%o A321813 (PARI) apply( A321813(n)=sigma(n>>valuation(n,2),9), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%o A321813 (Python)
%o A321813 from sympy import divisor_sigma
%o A321813 def A321813(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),9)) # _Chai Wah Wu_, Jul 16 2022
%Y A321813 Column k=9 of A285425.
%Y A321813 Cf. A050999, A051000, A051001, A051002, A321810 - A321816 (analog for 2nd .. 12th powers).
%Y A321813 Cf. A321543 - A321565, A321807 - A321836 for related sequences.
%Y A321813 Cf. A000265, A013668, A013957.
%K A321813 nonn,mult
%O A321813 1,3
%A A321813 _N. J. A. Sloane_, Nov 24 2018