A263649 a(n) is multiplicative with a(2^e) = (-1)^e, a(3^e) = -2*(-1)^e if e>0, a(p^e) = e+1 if p == 1, 7 (mod 24), a(p^e) = (e+1) * (-1)^e if p == 5, 11 (mod 24), a(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).

1, -1, 2, 1, -2, -2, 2, -1, -2, 2, -2, 2, 0, -2, -4, 1, 0, 2, 0, -2, 4, 2, 0, -2, 3, 0, 2, 2, -2, 4, 2, -1, -4, 0, -4, -2, 0, 0, 0, 2, 0, -4, 0, -2, 4, 0, 0, 2, 3, -3, 0, 0, -2, -2, 4, -2, 0, 2, -2, -4, 0, -2, -4, 1, 0, 4, 0, 0, 0, 4, 0, 2, 2, 0, 6, 0, -4, 0

Offset

1,3

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

a(2*n) = - a(n). a(3*n) = 2 * A115660(n). a(3*n + 1) = A263571(n+1). a(3*n + 2) = - A263548(n).

a(6*n + 1) = A261115(n). a(6*n + 3) = 2 * A128580(n). a(6*n + 5) = -2 * A128582(n).

Sum_{k=1..n} abs(a(k)) ~ (2/3)*sqrt(2/3)*Pi*n. - Amiram Eldar, Jan 29 2024

Example

G.f. = x - x^2 + 2*x^3 + x^4 - 2*x^5 - 2*x^6 + 2*x^7 - x^8 - 2*x^9 + ...

Mathematica

a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 4, {-1, 1, -2}[[#]] (-1)^#2, Mod[#, 24] < 12, (#2 + 1) KroneckerSymbol[ #, 12]^#2, True, 1 - Mod[#2, 2]]& @@@ FactorInteger[n])];

Prog

(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==2, (-1)^e, p==3, -2 * (-1)^e, p%24>12, 1-e%2, (e+1) * kronecker(p, 12)^e )))};

Crossrefs

Cf. A115660, A128580, A128582, A261115, A263548, A263571.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A083952 A245225 A214860 * A229904 A160242 A043529

Adjacent sequences: A263646 A263647 A263648 * A263650 A263651 A263652

Keyword

sign,mult

Author

Michael Somos, Oct 22 2015

Status

approved