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A321545 a(n) = Sum_{d|n} (-1)^(d-1)*d^6.

%I A321545 #19 Nov 04 2022 10:47:29
%S A321545 1,-63,730,-4159,15626,-45990,117650,-266303,532171,-984438,1771562,
%T A321545 -3036070,4826810,-7411950,11406980,-17043519,24137570,-33526773,
%U A321545 47045882,-64988534,85884500,-111608406,148035890,-194401190,244156251,-304089030,387952660,-489306350,594823322,-718639740,887503682
%N A321545 a(n) = Sum_{d|n} (-1)^(d-1)*d^6.
%H A321545 Seiichi Manyama, <a href="/A321545/b321545.txt">Table of n, a(n) for n = 1..10000</a>
%H A321545 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 A321545 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321545 G.f.: Sum_{k>=1} (-1)^(k-1)*k^6*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Dec 23 2018
%F A321545 Multiplicative with a(2^e) = 2 - (2^(6*e + 6) - 1)/63, and a(p^e) = (p^(6*e + 6) - 1)/(p^6 - 1) for p > 2. - _Amiram Eldar_, Nov 04 2022
%t A321545 f[p_, e_] := (p^(6*e + 6) - 1)/(p^6 - 1); f[2, e_] := 2 - (2^(6*e + 6) - 1)/63; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* _Amiram Eldar_, Nov 04 2022 *)
%o A321545 (PARI) apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^6), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%Y A321545 Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
%K A321545 sign,mult
%O A321545 1,2
%A A321545 _N. J. A. Sloane_, Nov 23 2018