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A321564 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^8.

%I A321564 #18 Nov 22 2022 22:00:42
%S A321564 1,-257,6562,-65281,390626,-1686434,5764802,-16711425,43053283,
%T A321564 -100390882,214358882,-428373922,815730722,-1481554114,2563287812,
%U A321564 -4278124289,6975757442,-11064693731,16983563042,-25500455906,37828630724,-55090232674
%N A321564 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^8.
%H A321564 Seiichi Manyama, <a href="/A321564/b321564.txt">Table of n, a(n) for n = 1..10000</a>
%H A321564 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 A321564 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321564 G.f.: Sum_{k>=1} (-1)^(k+1)*k^8*x^k/(1 + x^k). - _Ilya Gutkovskiy_, Dec 22 2018
%F A321564 Multiplicative with a(2^e) = -(127*2^(8*e+1) + 511)/255, and a(p^e) = (p^(8*e+8) - 1)/(p^8 - 1) for p > 2. - _Amiram Eldar_, Nov 22 2022
%t A321564 a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^8 &]; Array[a, 25] (* _Amiram Eldar_, Nov 22 2022 *)
%o A321564 (PARI) apply( A321564(n)=sumdiv(n, d, (-1)^(n\d-d)*d^8), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%Y A321564 Column k=8 of A322083.
%Y A321564 Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
%K A321564 sign,mult
%O A321564 1,2
%A A321564 _N. J. A. Sloane_, Nov 23 2018