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%I #61 Jan 23 2024 15:05:25
%S 1,-2,2,-1,2,-4,2,0,3,-4,2,-2,2,-4,4,1,2,-6,2,-2,4,-4,2,0,3,-4,4,-2,2,
%T -8,2,2,4,-4,4,-3,2,-4,4,0,2,-8,2,-2,6,-4,2,2,3,-6,4,-2,2,-8,4,0,4,-4,
%U 2,-4,2,-4,6,3,4,-8,2,-2,4,-8,2,0,2,-4,6,-2,4
%N G.f.: Sum_{k>0} -(-x)^k / (1 + x^k).
%H G. C. Greubel, <a href="/A228441/b228441.txt">Table of n, a(n) for n = 1..5000</a>
%H Peter Bala, <a href="/A067856/a067856_1.pdf">A signed Dirichlet product of arithmetical functions</a>, 2019.
%H Subhash Chand Bhoria, Pramod Eyyunni, and Bibekananda Maji, <a href="https://arxiv.org/abs/2011.07767">Generalization of five q-series identities of Ramanujan and unexplored weighted partition identities</a>, arXiv:2011.07767 [math.NT], 2020.
%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 a(n) = number of divisors of n minus 4 times number of divisors of n of the form 4*k+2.
%F a(n) = Sum_{d|n} (-1)^(d+n/d). - _N. J. A. Sloane_, Nov 23 2018
%F Multiplicative with a(2^e) = e-3 if e>0, a(p^e) = e+1 if p>2.
%F Moebius transform is period 4 sequence [1, -3, 1, 1, ...].
%F G.f.: Sum_{k>0} x^k / (1 - x^k) - 4 * x^(4*k + 2) / (1 - x^(4*k + 2)).
%F a(2*n - 1) = A099774(n).
%F Dirichlet g.f.: zeta(s)^2*(1-2^(-s+1))^2 = eta^2(s) (the Dirichlet eta). - _Ralf Stephan_, Mar 27 2015
%F a(16n+8) = a(A051062(n)) = 0. - _Michel Marcus_, Mar 27 2015
%F O.g.f.: Sum_{n >= 1} (-1)^(n*(n+1))*x^(n^2)*(1 - x^n)/(1 + x^n). - _Peter Bala_, Mar 11 2019
%F Conjecture: a(n) = (7 - 2*(-1)^n)*tau(n) - 4*tau(2*n) = 5*tau(n) - (3 + (-1)^n)*tau(2*n), where tau = A000005. - _Velin Yanev_, Dec 17 2019
%F The proof of the above conjecture easily follows from the fact that both a(n) and tau(n) are multiplicative arithmetical functions and tau(p^e) = e + 1 for prime p. - _Peter Bala_, Jan 28 2022
%F a(n) = A000005(n) if n is odd, and A000005(n) * (A007814(n)-3)/(A007814(n)+1) if n is even. - _Amiram Eldar_, Sep 18 2023
%e G.f. = x - 2*x^2 + 2*x^3 - x^4 + 2*x^5 - 4*x^6 + 2*x^7 + 3*x^9 - 4*x^10 + ...
%t a[ n_] := SeriesCoefficient[ Sum[ -(-x)^k / (1 + x^k), {k, 1, n}], {x, 0, n}];
%t a[ n_] := If[ n < 1, 0, DivisorSum[ n, (-1)^(# + n/#) &]]; (* _Michael Somos_, Jan 08 2015 *)
%t a[n_] := Module[{e = IntegerExponent[n, 2]}, DivisorSigma[0, n] * If[e == 0, 1, (e-3)/(e+1)]]; Array[a, 100] (* _Amiram Eldar_, Sep 18 2023 *)
%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, k, (-1)^(k + n/k)))};
%o (PARI) {a(n) = if( n<1, 0, numdiv(n) - 4 * sumdiv( n, k, k%4 == 2))};
%o (PARI) {a(n) = my(e); if( n<1, 0, e = valuation( n, 2); numdiv( n/2^e) * if( e>0, e-3, 1))};
%o (PARI) a(n)=direuler(p=1,n,if(p==2,(1-2*X)^2/(1-X)^2,1/(1-X)^2))[n] /* _Ralf Stephan_, Mar 27 2015 */
%Y Cf. A000005, A007814, A016825, A046897, A051062, A099774, A321558.
%K sign,easy,mult
%O 1,2
%A _Michael Somos_, Nov 02 2013
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