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A321812 Sum of 8th powers of odd divisors of n.

%I A321812 #24 Feb 16 2025 08:33:57
%S A321812 1,1,6562,1,390626,6562,5764802,1,43053283,390626,214358882,6562,
%T A321812 815730722,5764802,2563287812,1,6975757442,43053283,16983563042,
%U A321812 390626,37828630724,214358882,78310985282,6562,152588281251,815730722,282472589764
%N A321812 Sum of 8th powers of odd divisors of n.
%H A321812 Seiichi Manyama, <a href="/A321812/b321812.txt">Table of n, a(n) for n = 1..10000</a>
%H A321812 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 A321812 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OddDivisorFunction.html">Odd Divisor Function</a>.
%H A321812 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321812 a(n) = A013956(A000265(n)) = sigma_8(odd part of n); in particular, a(2^k) = 1 for all k >= 0. - _M. F. Hasler_, Nov 26 2018
%F A321812 G.f.: Sum_{k>=1} (2*k - 1)^8*x^(2*k-1)/(1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Dec 07 2018
%F A321812 From _Amiram Eldar_, Nov 02 2022: (Start)
%F A321812 Multiplicative with a(2^e) = 1 and a(p^e) = (p^(8*e+8)-1)/(p^8-1) for p > 2.
%F A321812 Sum_{k=1..n} a(k) ~ c * n^9, where c = zeta(9)/18 = 0.0556671... . (End)
%t A321812 a[n_] := DivisorSum[n, #^8 &, OddQ[#] &]; Array[a, 20] (* _Amiram Eldar_, Dec 07 2018 *)
%o A321812 (PARI) apply( A321812(n)=sigma(n>>valuation(n,2),8), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%o A321812 (Python)
%o A321812 from sympy import divisor_sigma
%o A321812 def A321812(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),8)) # _Chai Wah Wu_, Jul 16 2022
%Y A321812 Column k=8 of A285425.
%Y A321812 Cf. A050999, A051000, A051001, A051002, A321810 - A321816 (analog for 2nd .. 12th powers).
%Y A321812 Cf. A321543 - A321565, A321807 - A321836 for related sequences.
%Y A321812 Cf. A000265, A013667, A013956.
%K A321812 nonn,mult
%O A321812 1,3
%A A321812 _N. J. A. Sloane_, Nov 24 2018