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A002325 Glaisher's J numbers.
(Formerly M0043 N0013)
40
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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.

For n nonzero, a(n) is nonzero if and only if n is in A002479. - Michael Somos, Dec 15 2011

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

REFERENCES

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 114 Entry 8(iii).

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.

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.24).

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. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

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. [Incomplete annotated scanned copy]

Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.

N. J. A. Sloane, Transforms.

Index entries for sequences related to Glaisher's numbers.

FORMULA

Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m, p)+1)*p^(-s) + Kronecker(m, p)*p^(-2s))^(-1) for m = -2.

Moebius transform is period 8 sequence [ 1, 0, 1, 0, -1, 0, -1, 0, ...]. - Michael Somos, Aug 23 2005

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)).

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

A033715(n) = 2 * a(n) unless n=0.

a(n) = A188169(n) + A188170(n) - A188171(n) - A188172(n) [Hirschhorn]. - R. J. Mathar, Mar 23 2011

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

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

EXAMPLE

x + x^2 + 2*x^3 + x^4 + 2*x^6 + x^8 + 3*x^9 + 2*x^11 + 2*x^12 + x^16 + ...

MAPLE

S:= series( (JacobiTheta3(0, q)*JacobiTheta3(0, q^2)-1)/2, q, 1001):

seq(coeff(S, q, j), j=1..1000); # Robert Israel, Dec 01 2015

MATHEMATICA

a[n_] := Total[ KroneckerSymbol[-8, #] & /@ Divisors[n]]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Nov 25 2011, after Michael Somos *)

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 *)

PROG

(PARI) a(n) = if( n<1, 0, issquare(n)-issquare(2*n) + 2*sum(i=1, sqrtint(n\2), issquare(n-2*i^2)))

(PARI) {a(n) = if( n<1, 0, qfrep([ 1, 0; 0, 2], n)[n])} \\ Michael Somos, Jun 05 2005

(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

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -2, d)))} \\ Michael Somos, Aug 23 2005

(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

(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)}

(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

CROSSREFS

Cf. A033715, A093954.

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.

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.

Sequence in context: A220136 A322454 A133693 * A129134 A065675 A348128

Adjacent sequences: A002322 A002323 A002324 * A002326 A002327 A002328

KEYWORD

nonn,easy,nice,mult

AUTHOR

N. J. A. Sloane

STATUS

approved

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