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%I #47 Nov 16 2023 07:44:26
%S 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,
%T 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,
%U 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
%N a(n) = Sum_{d|n} Kronecker(-12, d) (= A134667(d)).
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 346.
%H G. C. Greubel, <a href="/A035178/b035178.txt">Table of n, a(n) for n = 1..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F Moebius transform is period 6 sequence [ 1, 0, 0, 0, -1, 0, ...]. - _Michael Somos_, Feb 14 2006
%F 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
%F 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
%F 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).
%F 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
%F a(n) = |A093829(n)| = -(-1)^n * A137608(n) = a(2*n) = a(3*n). a(6*n + 1) = A097195(n). a(6*n + 5) = 0.
%F From _Michael Somos_, Aug 11 2009: (Start)
%F 3 * a(n) = A107760(n) unless n=0. a(2*n + 1) = A033762(n). a(3*n + 1) = A033687(n). a(4*n + 1) = A112604(n). a(4*n + 3) = A112605(n).
%F a(8*n + 1) = A112606(n). a(8*n + 3) = A112608(n). a(8*n + 5) = 2 * A112607(n). a(8*n + 7) = 2 * A112608(n). a(12*n + 1) A123884(n). a(12*n + 7) = 2 * A121361(n).
%F a(24*n + 1) = A131961(n). a(24*n + 7) = 2 * A131962(n). a(24*n + 13) = 2 * A131963(n). a(24*n + 19) = 2 * A131964(n). (End)
%F 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
%F 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
%e 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 + ...
%t a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -12, d], { d, Divisors[ n]}]]; (* _Michael Somos_, Jun 24 2011 *)
%t 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 *)
%t 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 *)
%t a[n_] := DivisorSum[n, KroneckerSymbol[-12, #]&]; Array[a, 105] (* _Jean-François Alcover_, Dec 01 2015 *)
%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -12, d)))}; /* _Michael Somos_, Apr 18 2004 */
%o (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 */
%o (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 */
%o (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 */
%o (Magma) A := Basis( ModularForms( Gamma1(6), 1), 88); B<q> := (A[1] - 1) / 3 + A[2]; B; /* _Michael Somos_, Aug 04 2015 */
%Y Cf. A033687, A033762, A093829, A093766, A097197, A107760, A112604, A112605.
%Y Cf. A112606, A112607, A112608, A121361, A123884, A131961, A131962.
%Y Cf. A131963, A131964, A137608.
%Y Cf. A000122, A000700, A010054, A121373.
%K nonn,easy,mult
%O 1,7
%A _N. J. A. Sloane_
%E Definition edited by _Michael Somos_, Aug 11 2009
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