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A321563 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^7.

%I A321563 #18 Nov 22 2022 22:00:32
%S A321563 1,-129,2188,-16257,78126,-282252,823544,-2080641,4785157,-10078254,
%T A321563 19487172,-35570316,62748518,-106237176,170939688,-266321793,
%U A321563 410338674,-617285253,893871740,-1270094382,1801914272,-2513845188,3404825448,-4552442508
%N A321563 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^7.
%H A321563 Seiichi Manyama, <a href="/A321563/b321563.txt">Table of n, a(n) for n = 1..10000</a>
%H A321563 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 A321563 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321563 G.f.: Sum_{k>=1} (-1)^(k+1)*k^7*x^k/(1 + x^k). - _Ilya Gutkovskiy_, Dec 22 2018
%F A321563 Multiplicative with a(2^e) = -3*(21*2^(7*e+1) + 85)/127, and a(p^e) = (p^(7*e+7) - 1)/(p^7 - 1) for p > 2. - _Amiram Eldar_, Nov 22 2022
%t A321563 a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^7 &]; Array[a, 25] (* _Amiram Eldar_, Nov 22 2022 *)
%o A321563 (PARI) apply( A321563(n)=sumdiv(n, d, (-1)^(n\d-d)*d^7), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%Y A321563 Column k=7 of A322083.
%Y A321563 Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
%K A321563 sign,mult
%O A321563 1,2
%A A321563 _N. J. A. Sloane_, Nov 23 2018