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A035185
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Number of divisors of n == 1 or 7 (mod 8) minus number of divisors of n == 3 or 5 (mod 8).
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2
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1, 1, 0, 1, 0, 0, 2, 1, 1, 0, 0, 0, 0, 2, 0, 1, 2, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 2, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 3, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 2, 1, 0, 0, 0, 2, 0, 0, 2, 1, 2, 0, 0, 0, 0, 0, 2, 0, 1, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 2, 3, 0, 1, 0, 0, 2, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,7
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COMMENTS
| Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 2.
Let zetaQ(sqrt(2))(s)=sum(1/(Z(sqrt(2)):A)^s), a Dedekind zeta function, where A runs through the nonzero ideals of Z(sqrt(2)) and where (Z(sqrt(2)):A) is the norm of A; then zetaQ(sqrt(2))(s)=sum(n>=1, a(n)/n^s); sum(k=1, n, a(k)) is asymptotic to c*n where c=log(1+sqrt(2))/sqrt(2)
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LINKS
| M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. 51 (1999), 1258-1276.
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FORMULA
| G.f.: Sum_{n>0} x^n*(1-x^(2*n))/(1+x^(4*n)).
-(-1)^(n*(n-1)/2)*a(n) = Sum_{n >= 1} (-1)^n * q^(n*(n+1)/2)*(1-q)*(1-q^2)*...*(1-q^(n-1))/ ((1+q)*(1+q^2)*...*(1+q^n)). [From Jeremy Lovejoy, Jun 12 2009]
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PROG
| (PARI) a(n)=if(n<1, 0, sumdiv(n, d, kronecker(2, d)))
(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-X)/(1-kronecker(2, p)*X))[n])
(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>1&p%8<7, !(e%2), e+1)))))} /* Michael Somos Aug 17 2006 */
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CROSSREFS
| Cf. A035251, A078462, A188169, A188170, A188171, A188172
Sequence in context: A025923 A138158 A057276 * A086013 A167687 A064692
Adjacent sequences: A035182 A035183 A035184 * A035186 A035187 A035188
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KEYWORD
| nonn,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Additional comments from Benoit Cloitre, Jan 01, 2003
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