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A035178
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a(n) = sum_{d|n} kronecker( -12, d) (= A134667(d)).
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5
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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; refs; listen; history; internal format)
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OFFSET
| 1,7
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FORMULA
| Moebius transform is period 6 sequence [ 1, 0, 0, 0, -1, 0, ...]. - Michael Somos Feb 14 2006
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
Dirichlet g.f.: zeta(s) * L(chi,s) where chi(n) = kronecker( -12, n). Sum_{n>0} a(n) / n^s = Product_{p prime} 1 / ((1 - p^-s) * (1 - kronecker( -12, p) * p^-s)). - Michael Somos Jun 24 2011 */
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).
G.f.: Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(2*k) + x^(4*k)) = Sum_{k>=0} x^(6*k + 1) / (1 - x^(6*k + 1)) -x^(6*k + 5) /( 1 - x^(6*k + 5)). - Michael Somos Feb 14 2006
a(n) = |A093829(n)|. A097195(n)= a(6*n + 1). A033687(n)= a(6*n + 2). A033762(n) = a(6*n + 3). a(6*n + 5) = 0.
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
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EXAMPLE
| q + q^2 + q^3 + q^4 + q^6 + 2*q^7 + q^8 + q^9 + q^12 + 2*q^13 + 2*q^14 + ...
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MATHEMATICA
| a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -12, d], { d, Divisors[ n]}]] (* Michael Somos Jun 24 2011 *)
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PROG
| (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 Jun 24 2011 */
(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
| Cf. A033687, A033762, A093829, A097197, A107760, A112604, A112605, A112606, A112607, A112608, A121361, A123884, A131961, A131962, A131963, A131964.
Sequence in context: * A093829 A113447 A137608 A191336 A078807 A029422
Adjacent sequences: A035175 A035176 A035177 * A035179 A035180 A035181
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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
| Definition edited by Michael Somos Aug 11 2009
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