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Search: id:A035178
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| A035178 |
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Sum over divisors d of n of Kronecker symbol (-12,d) (= A134667(d)). |
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+0
6
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| 1, 1, 1, 1, 0, 1, 2, 1, 1, 0, 0, 1, 2, 2, 0, 1, 0, 1, 2, 0, 2, 0, 0, 1, 1, 2, 1, 2, 0, 0, 2, 1, 0, 0, 0, 1, 2, 2, 2, 0, 0, 2, 2, 0, 0, 0, 0, 1, 3, 1, 0, 2, 0, 1, 0, 2, 2, 0, 0, 0, 2, 2, 2, 1, 0, 0, 2, 0, 0, 0, 0, 1, 2, 2, 1, 2, 0, 2, 2, 0, 1, 0, 0, 2, 0, 2, 0, 0, 0, 0, 4, 0, 2, 0, 0, 1, 2, 3, 0, 1, 0, 0, 2, 2, 0
(list; graph; listen)
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OFFSET
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1,7
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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 = -12.
G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=(u1-u2)*(u1-u2-u3+u6)-(u2-u6)*(1+3*u6) . - Michael Somos May 29 2005
Multiplicative with a(p^e) = 1 if p=2 or p=3; a(p^e) = 1+e if p == 1 (mod 6); a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
Moebius transform is period 6 sequence [1,0,0,0,-1,0,...]. - Michael Somos Feb 14 2006
G.f.: Sum_{k>0} (x^k+x^(3k))/(1+x^(2k)+x^(4k)) = Sum_{k>=0} x^(6k+1)/(1-x^(6k+1)) -x^(6k+5)/(1-x^(6k+5)) . - Michael Somos Feb 14 2006
a(6n+5) = 0.
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EXAMPLE
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q + q^2 + q^3 + q^4 + q^6 + 2*q^7 + q^8 + q^9 + q^12 + 2*q^13 + 2*q^14 + ... - Michael Somos Aug 11 2009
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PROGRAM
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(PARI) a(n)=if(n<1, 0, sumdiv(n, d, kronecker(-12, d))) /* Michael Somos Apr 18 2004 */
(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-X)/(1-kronecker(-12, p)*X))[n]) /* Michael Somos Apr 18 2004 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^2 + A)^6 / (eta(x^6 + A)^2 * eta(x + A)^3) - 1) / 3, n))} /* Michael Somos Aug 11 2009 */ - Michael Somos Aug 11 2009
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CROSSREFS
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|A093829(n)|=a(n). A097195(n)= a(6n+1). A033687(n)= a(6n+2). A033762(n)= a(6n+3).
A107760(n) = 3*a(n) unless n=0. A033762(n) = a(2*n + 1). A033687(n) = a(3*n + 1). A112604(n) = a(4*n + 1). A112605(n) = a(4*n + 3). - Michael Somos Aug 11 2009
A112606(n) = a(8*n + 1). A112608(n) = a(8*n + 3). 2 * A112607(n) = a(8*n + 5). 2 * A112608(n) = a(8*n + 7). A123884(n) = a(12*n + 1). 2 * A121361(n) = a(12*n + 7). - Michael Somos Aug 11 2009
A131961(n) = a(24*n + 1). 2 * A131962(n) = a(24*n + 7). 2 * A131963(n) = a(24*n + 13). 2 * A131964(n) = a(24*n + 19). - Michael Somos Aug 11 2009
Sequence in context: A134404 A107110 A061197 this_sequence A093829 A113447 A137608
Adjacent sequences: A035175 A035176 A035177 this_sequence A035179 A035180 A035181
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KEYWORD
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nonn,mult
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Definition edited by Michael Somos Aug 11 2009
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