cp's OEIS Frontend

A002325 Glaisher's J numbers.

%I A002325 M0043 N0013 #76 Jul 25 2025 15:41:28
%S A002325 1,1,2,1,0,2,0,1,3,0,2,2,0,0,0,1,2,3,2,0,0,2,0,2,1,0,4,0,0,0,0,1,4,2,
%T A002325 0,3,0,2,0,0,2,0,2,2,0,0,0,2,1,1,4,0,0,4,0,0,4,0,2,0,0,0,0,1,0,4,2,2,
%U A002325 0,0,0,3,2,0,2,2,0,0,0,0,5,2,2,0,0,2,0,2,2,0,0,0,0,0,0,2,2,1,6,1,0,4,0,0,0
%N A002325 Glaisher's J numbers.
%C A002325 Number of integer solutions to the equation x^2 + 2*y^2 = n when (-x, -y) and (x, y) are counted as the same solution.
%C A002325 For n nonzero, a(n) is nonzero if and only if n is in A002479. - _Michael Somos_, Dec 15 2011
%C A002325 Coefficients of Dedekind zeta function for the quadratic number field of discriminant -8. See A002324 for formula and Maple code. - _N. J. A. Sloane_, Mar 22 2022
%D A002325 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 114 Entry 8(iii).
%D A002325 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 19.
%D A002325 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.24).
%D A002325 J. W. L. Glaisher, Table of the excess of the number of (8k+1)- and (8k+3)-divisors of a number over the number of (8k+5)- and (8k+7)-divisors, Messenger Math., 31 (1901), 82-91.
%D A002325 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
%D A002325 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002325 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002325 T. D. Noe, <a href="/A002325/b002325.txt">Table of n, a(n) for n = 1..10000</a>
%H A002325 J. W. L. Glaisher, <a href="/A002325/a002325.pdf">Table of the excess of the number of (8k+1)- and (8k+3)-divisors of a number over the number of (8k+5)- and (8k+7)-divisors</a>, Messenger Math., 31 (1901), 82-91. [Incomplete annotated scanned copy]
%H A002325 Michael D. Hirschhorn, <a href="http://dx.doi.org/10.1016/j.disc.2004.08.045">The number of representations of a number by various forms</a>, Discrete Mathematics 298 (2005), 205-211.
%H A002325 Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/2507.15780">Pairs of intertwined integer sequences</a>, arXiv:2507.15780 [math.NT], 2025. See p. 5.
%H A002325 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%H A002325 <a href="/index/Ge#Glaisher">Index entries for sequences related to Glaisher's numbers</a>.
%F A002325 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m, p)+1)*p^(-s) + Kronecker(m, p)*p^(-2s))^(-1) for m = -2.
%F A002325 Moebius transform is period 8 sequence [ 1, 0, 1, 0, -1, 0, -1, 0, ...]. - _Michael Somos_, Aug 23 2005
%F A002325 G.f.: (theta_3(q) * theta_3(q^2) - 1) / 2 = Sum_{k>0} Kronecker( -2, n) * x^k / (1 - x^k) = Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(4*k)).
%F A002325 Multiplicative with a(2^e) = 1, a(p^e) = e+1 if p == 1, 3 (mod 8), a(p^e) = (1+(-1)^e)/2 if p == 5, 7 (mod 8). - _Michael Somos_, Oct 23 2006
%F A002325 A033715(n) = 2 * a(n) unless n=0.
%F A002325 a(n) = A188169(n) + A188170(n) - A188171(n) - A188172(n) [Hirschhorn]. - _R. J. Mathar_, Mar 23 2011
%F A002325 G.f.: A(x) = 2*(1+x^2)/(G(0)-2*x*(1+x^2)); G(k) = 1+x+x^(2*k)*(1+x^3+x^(2*k+1)+x^(2*k+4)+x^(4*k+3)+x^(4*k+4)) - x*(1+x^(2*k))*(1+x^(2*k+4))*(1+x^(4*k+4))^2/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jan 03 2012
%F A002325 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(2)) = 1.110720... (A093954). - _Amiram Eldar_, Oct 11 2022
%e A002325 x + x^2 + 2*x^3 + x^4 + 2*x^6 + x^8 + 3*x^9 + 2*x^11 + 2*x^12 + x^16 + ...
%p A002325 S:= series( (JacobiTheta3(0,q)*JacobiTheta3(0,q^2)-1)/2, q, 1001):
%p A002325 seq(coeff(S,q,j), j=1..1000); # _Robert Israel_, Dec 01 2015
%t A002325 a[n_] := Total[ KroneckerSymbol[-8, #] & /@ Divisors[n]]; Table[a[n], {n, 1, 105}] (* _Jean-François Alcover_, Nov 25 2011, after _Michael Somos_ *)
%t A002325 QP = QPochhammer; s = ((QP[q^2]^3*QP[q^4]^3)/(QP[q]^2*QP[q^8]^2)-1)/(2q) + O[q]^105; CoefficientList[s, q] (* _Jean-François Alcover_, Dec 01 2015, adapted from PARI *)
%o A002325 (PARI) a(n) = if( n<1, 0, issquare(n)-issquare(2*n) + 2*sum(i=1,sqrtint(n\2), issquare(n-2*i^2)))
%o A002325 (PARI) {a(n) = if( n<1, 0, qfrep([ 1, 0; 0, 2],n)[n])} \\ _Michael Somos_, Jun 05 2005
%o A002325 (PARI) {a(n) = if( n<1, 0, direuler(p=2, n, 1 / (1 - X) / (1 - kronecker( -2, p) * X))[n])} \\ _Michael Somos_, Jun 05 2005
%o A002325 (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -2, d)))} \\ _Michael Somos_, Aug 23 2005
%o A002325 (PARI) {a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 1, if( p%8<4, e+1, !(e%2))))))} \\ _Michael Somos_, Oct 23 2006
%o A002325 (PARI) {a(n) = local(A); if( n<1, 0, A = x * O(x^n); polcoeff( eta(x + A)^-2 * eta(x^2 + A)^3 * eta(x^4 + A)^3 * eta(x^8 + A)^-2, n) / 2)}
%o A002325 (PARI) a(n) = my(f=factor(n>>valuation(n,2)), e); prod(i=1, #f~, e=f[i, 2]; if( f[i, 1]%8<4, e+1, 1 - e%2)) \\ _Charles R Greathouse IV_, Sep 09 2014
%Y A002325 Cf. A033715, A093954.
%Y A002325 Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
%Y A002325 Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
%K A002325 nonn,easy,nice,mult
%O A002325 1,3
%A A002325 _N. J. A. Sloane_