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A321562 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^6.

%I A321562 #19 Nov 22 2022 22:00:24
%S A321562 1,-65,730,-4033,15626,-47450,117650,-257985,532171,-1015690,1771562,
%T A321562 -2944090,4826810,-7647250,11406980,-16510913,24137570,-34591115,
%U A321562 47045882,-63019658,85884500,-115151530,148035890,-188329050,244156251,-313742650
%N A321562 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^6.
%H A321562 G. C. Greubel, <a href="/A321562/b321562.txt">Table of n, a(n) for n = 1..1000</a>
%H A321562 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 A321562 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321562 G.f.: Sum_{k>=1} (-1)^(k+1)*k^6*x^k/(1 + x^k). - _Ilya Gutkovskiy_, Nov 27 2018
%F A321562 Multiplicative with a(2^e) = -(31*2^(6*e+1) + 127)/63, and a(p^e) = (p^(6*e+6) - 1)/(p^6 - 1) for p > 2. - _Amiram Eldar_, Nov 22 2022
%t A321562 a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^6 &]; Array[a, 50] (* _Amiram Eldar_, Nov 27 2018 *)
%o A321562 (PARI) apply( A321562(n)=sumdiv(n, d, (-1)^(n\d-d)*d^6), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%o A321562 (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(-1)^(k+1)*k^6*x^k/(1 + x^k) : k in [1..2*m]]) )); // _G. C. Greubel_, Nov 28 2018
%o A321562 (Sage) s=(sum((-1)^(k+1)*k^6*x^k/(1 + x^k)  for k in (1..50))).series(x, 50); a = s.coefficients(x, sparse=False); a[1:] # _G. C. Greubel_, Nov 28 2018
%Y A321562 Column k=6 of A322083.
%Y A321562 Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
%K A321562 sign,mult
%O A321562 1,2
%A A321562 _N. J. A. Sloane_, Nov 23 2018