Search: id:A002325
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%I A002325 M0043 N0013
%S A002325 1,1,2,1,0,2,0,1,3,0,2,2,0,0,0,1,2,3,2,0,0,2,0,2,1,0,4,0,0,0,0,1,4,2,0,
%T A002325 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,
%U A002325 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+2y^2=n when (-x,-y) and 
               (x,y) are counted as the same solution.
%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="b002325.txt">Table of n, a(n) for n=1..10000</a>
%H A002325 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%H A002325 <a href="Sindx_Ge.html#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(-8, n)x^k/(1-x^k) 
               = Sum_{k>0} (x^k+x^(3k))/(1+x^(4k)).
%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
%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(-8,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)}
%Y A002325 Cf. A033715(n) = 2 * a(n) unless n=0.
%Y A002325 Sequence in context: A080884 A091392 A036577 this_sequence A129134 A133693 
               A065675
%Y A002325 Adjacent sequences: A002322 A002323 A002324 this_sequence A002326 A002327 
               A002328
%K A002325 nonn,easy,nice,mult
%O A002325 1,3
%A A002325 N. J. A. Sloane (njas(AT)research.att.com).

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