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%I #24 Nov 04 2022 10:47:14
%S 1,-3,10,-19,26,-30,50,-83,91,-78,122,-190,170,-150,260,-339,290,-273,
%T 362,-494,500,-366,530,-830,651,-510,820,-950,842,-780,962,-1363,1220,
%U -870,1300,-1729,1370,-1086,1700,-2158,1682,-1500,1850,-2318,2366,-1590,2210,-3390,2451,-1953,2900,-3230,2810,-2460,3172
%N a(n) = Sum_{d|n} (-1)^(d-1)*d^2.
%H Seiichi Manyama, <a href="/A321543/b321543.txt">Table of n, a(n) for n = 1..10000</a>
%H J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&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 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F G.f.: Sum_{k>=1} (-1)^(k-1)*k^2*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Dec 23 2018
%F G.f.: Sum_{n >= 1} x^n*(1 - x^n)/(1 + x^n)^3. - _Peter Bala_, Jan 11 2021
%F Multiplicative with a(2^e) = 2 - (2^(2*e + 2) - 1)/3, and a(p^e) = (p^(2*e + 2) - 1)/(p^2 - 1) for p > 2. - _Amiram Eldar_, Nov 04 2022
%p with(numtheory):
%p a := n -> add( (-1)^(d-1)*d^2, d in divisors(n) ): seq(a(n), n = 1..40);
%p # _Peter Bala_, Jan 11 2021
%t f[p_, e_] := (p^(2*e + 2) - 1)/(p^2 - 1); f[2, e_] := 2 - (2^(2*e + 2) - 1)/3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 55] (* _Amiram Eldar_, Nov 04 2022 *)
%o (PARI) apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^2), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%Y Apart from signs, same as A064027.
%Y Glaisher's zeta_i, i=0..12: A048272, A002129, A321543, A138503, A279395, A321544, A321545, A321546, A321547, A321548, A321549, A321550, A321551.
%Y Cf. A321552 - A321565, A321807 - A321836 for similar sequences.
%K sign,mult
%O 1,2
%A _N. J. A. Sloane_, Nov 23 2018
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