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Search: id:A002325
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| A002325 |
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Glaisher's J numbers.
(Formerly M0043 N0013)
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+0
8
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| 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; listen)
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OFFSET
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1,3
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COMMENT
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Number of integer solutions to the equation x^2+2y^2=n when (-x,-y) and (x,y) are counted as the same solution.
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REFERENCES
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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).
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..10000
N. J. A. Sloane, Transforms
Index entries for sequences related to Glaisher's numbers
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FORMULA
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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(-8, n)x^k/(1-x^k) = Sum_{k>0} (x^k+x^(3k))/(1+x^(4k)).
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
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PROGRAM
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(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(-8, 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)}
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CROSSREFS
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Cf. A033715(n) = 2 * a(n) unless n=0.
Sequence in context: A080884 A091392 A036577 this_sequence A129134 A133693 A065675
Adjacent sequences: A002322 A002323 A002324 this_sequence A002326 A002327 A002328
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KEYWORD
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nonn,easy,nice,mult
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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