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A321549 a(n) = Sum_{d|n} (-1)^(d-1)*d^10.

%I A321549 #18 Nov 04 2022 10:47:49
%S A321549 1,-1023,59050,-1049599,9765626,-60408150,282475250,-1074791423,
%T A321549 3486843451,-9990235398,25937424602,-61978820950,137858491850,
%U A321549 -288972180750,576660215300,-1100586419199,2015993900450,-3567040850373,6131066257802,-10249991283974,16680163512500,-26533985367846,41426511213650
%N A321549 a(n) = Sum_{d|n} (-1)^(d-1)*d^10.
%H A321549 Seiichi Manyama, <a href="/A321549/b321549.txt">Table of n, a(n) for n = 1..10000</a>
%H A321549 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 A321549 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321549 G.f.: Sum_{k>=1} (-1)^(k-1)*k^10*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Dec 23 2018
%F A321549 Multiplicative with a(2^e) = 2 - (2^(10*e + 10) - 1)/1023, and a(p^e) = (p^(10*e + 10) - 1)/(p^10 - 1) for p > 2. - _Amiram Eldar_, Nov 04 2022
%t A321549 f[p_, e_] := (p^(10*e + 10) - 1)/(p^10 - 1); f[2, e_] := 2 - (2^(10*e + 10) - 1)/1023; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* _Amiram Eldar_, Nov 04 2022 *)
%o A321549 (PARI) apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^10), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%Y A321549 Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
%K A321549 sign,mult
%O A321549 1,2
%A A321549 _N. J. A. Sloane_, Nov 23 2018