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A035178
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a(n) = Sum_{d|n} Kronecker(-12, d) (= A134667(d)).
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10
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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
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OFFSET
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1,7
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COMMENTS
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REFERENCES
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J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 346.
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LINKS
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FORMULA
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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
a(n) is 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
Expansion of (psi(q)^3 / psi(q^3) - 1) / 3 in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Aug 04 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Nov 16 2023
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EXAMPLE
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G.f. = 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
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a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -12, d], { d, Divisors[ n]}]]; (* Michael Somos, Jun 24 2011 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 5, 1, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1 ] & @@@ FactorInteger@n)]; (* Michael Somos, Aug 04 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^(1/2)]^3 / EllipticTheta[ 2, 0, q^(3/2)] - 4) / 12, {q, 0, n}]; (* Michael Somos, Aug 04 2015 *)
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PROG
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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, Jun 24 2011 */
(PARI) {a(n) = my(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 */
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 1, p%6==5, 1-e%2, 1+e)))}; /* Michael Somos, Aug 04 2015 */
(Magma) A := Basis( ModularForms( Gamma1(6), 1), 88); B<q> := (A[1] - 1) / 3 + A[2]; B; /* Michael Somos, Aug 04 2015 */
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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