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A321818 a(n) = Sum_{d|n, n/d odd} d^8 for n > 0.

%I A321818 #26 Jan 09 2023 01:50:25
%S A321818 1,256,6562,65536,390626,1679872,5764802,16777216,43053283,100000256,
%T A321818 214358882,430047232,815730722,1475789312,2563287812,4294967296,
%U A321818 6975757442,11021640448,16983563042,25600065536,37828630724,54875873792,78310985282
%N A321818 a(n) = Sum_{d|n, n/d odd} d^8 for n > 0.
%H A321818 Seiichi Manyama, <a href="/A321818/b321818.txt">Table of n, a(n) for n = 1..10000</a>
%H A321818 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 A321818 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321818 G.f.: Sum_{k>=1} k^8*x^k/(1 - x^(2*k)). - _Ilya Gutkovskiy_, Dec 22 2018
%F A321818 From _Amiram Eldar_, Nov 02 2022: (Start)
%F A321818 Multiplicative with a(2^e) = 2^(8*e) and a(p^e) = (p^(8*e+8)-1)/(p^8-1) for p > 2.
%F A321818 Sum_{k=1..n} a(k) ~ c * n^9, where c = 511*zeta(9)/4608 = 0.1111168... . (End)
%F A321818 Dirichlet g.f.: zeta(s)*zeta(s-8)*(1-1/2^s). - _Amiram Eldar_, Jan 09 2023
%t A321818 a[n_] := DivisorSum[n, #^8 &, OddQ[n/#] &]; Array[a, 24] (* _Amiram Eldar_, Nov 02 2022 *)
%o A321818 (PARI) apply( A321818(n)=sumdiv(n,d,if(bittest(n\d,0),d^8)), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%Y A321818 Cf. A321543 - A321565, A321807 - A321836 for related sequences.
%Y A321818 Cf. A013667.
%K A321818 nonn,mult
%O A321818 1,2
%A A321818 _N. J. A. Sloane_, Nov 24 2018