A192013 a(n) = Sum_{d|n} Kronecker(-6, d).

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 0, 2, 2, 1, 0, 1, 0, 2, 2, 2, 0, 1, 3, 0, 1, 2, 2, 2, 2, 1, 2, 0, 4, 1, 0, 0, 0, 2, 0, 2, 0, 2, 2, 0, 0, 1, 3, 3, 0, 0, 2, 1, 4, 2, 0, 2, 2, 2, 0, 2, 2, 1, 0, 2, 0, 0, 0, 4, 0, 1, 2, 0, 3, 0, 4, 0, 2, 2, 1, 0, 2, 2, 0, 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 1, 2, 3, 2, 3, 2, 0, 2, 0, 4

Offset

1,5

Links

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

Formula

Moebius transform is period 24 sequence [ 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, ...].

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

G.f.: Sum_{k>0} x^k * (1 + x^(4*k)) * (1 + x^(6*k)) / (1 + x^(12*k)).

Dirichlet g.f.: zeta(s) * L(chi,s) where chi(n) = Kronecker(-6, n). Sum_{n>0} a(n) / n^s = Product_{p prime} 1 / ((1 - p^-s) * (1 - Kronecker(-6, p) * p^-s)).

a(n) = A000377(n) = A000377(2*n) = A190611(2*n). a(n) = a(2*n) = a(3*n). - Michael Somos, Jul 22 2015

0 <= a(n) <= d(n) and these bounds are sharp. - Charles R Greathouse IV, Dec 14 2016

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(6) = 1.282549... . - Amiram Eldar, Oct 17 2022

Example

G.f. = x + x^2 + x^3 + x^4 + 2*x^5 + x^6 + 2*x^7 + x^8 + x^9 + 2*x^10 + ...

Mathematica

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -6, d], { d, Divisors[n]}]];

Prog

(PARI) {a(n) = sumdiv( n, d, kronecker( -6, d))};
(PARI) a(n)=sumdivmult(n, d, kronecker(-6, d)) \\ Charles R Greathouse IV, Dec 14 2016

Crossrefs

Cf. A000377, A190611.

Cf. A109017(n) = Kronecker(-6, n). - Michael Somos, Jul 22 2015

Sequence in context: A128581 A190611 A000377 * A329618 A026517 A072047

Adjacent sequences: A192010 A192011 A192012 * A192014 A192015 A192016

Keyword

nonn,mult

Author

Michael Somos, Jun 22 2011

Status

approved