A065333 Characteristic function of 3-smooth numbers, i.e., numbers of the form 2^i*3^j (i, j >= 0).

1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Reinhard Zumkeller, Oct 29 2001

Keywords

• mult
• nonn
• easy

Comments

Dirichlet inverse of b(n) where b(n) = 0 except for: b(1) = b(6) = -b(2) = -b(3) = 1. - Alexander Adam, Dec 26 2012

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• A. Pakapongpun and T. Ward, Functorial Orbit counting, JIS 12 (2009) 09.2.4, example 9.
• Index entries for characteristic functions

Crossrefs

Characteristic function of A003586.

Cf. A000265, A007814, A007949, A038502, A065330, A065332, A071521 (partial sums), A072078 (inverse Möbius transform).

Programs

Haskell

a065333 = fromEnum . (== 1) . a038502 . a000265
-- Reinhard Zumkeller, Jan 08 2013, Apr 12 2012

Mathematica

a[n_] := Boole[ 2^IntegerExponent[n, 2] * 3^IntegerExponent[n, 3] == n]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, May 16 2013, after Charles R Greathouse IV *)

PARI

a(n)=sumdiv(n,d,moebius(6*d)) \\ Benoit Cloitre, Oct 18 2009

PARI

a(n)=3^valuation(n,3)<Charles R Greathouse IV, Aug 21 2011

Python

from sympy import multiplicity
def A065333(n): return int(3**(multiplicity(3,m:=n>>(~n&n-1).bit_length()))==m) # Chai Wah Wu, Dec 20 2024

Formula

a(n) = if n = A003586(k) for some k then 1 else 0.

a(n) = signum(A065332(n)), where signum = A057427.

a(n) = if A065330(n) = 1 then 1 else 0 = 1 - signum(A065330(n) - 1).

a(n) = Product_{p prime and p|n} 0^floor(p/4). - Reinhard Zumkeller, Nov 19 2004

Multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = 0 for prime p > 3. Dirichlet g.f. 1/(1-2^-s)/(1-3^-s). - Franklin T. Adams-Watters, Sep 01 2006

a(n) = 0^(A038502(A000265(n)) - 1). - Reinhard Zumkeller, Sep 28 2008

a(n) = Sum_{d|n} mu(6*d). - Benoit Cloitre, Oct 18 2009