A022329 Exponent of 3 (value of j) in n-th number of form 2^i*3^j (see A003586).

0, 0, 1, 0, 1, 0, 2, 1, 0, 2, 1, 3, 0, 2, 1, 3, 0, 2, 4, 1, 3, 0, 2, 4, 1, 3, 5, 0, 2, 4, 1, 3, 5, 0, 2, 4, 6, 1, 3, 5, 0, 2, 4, 6, 1, 3, 5, 0, 7, 2, 4, 6, 1, 3, 5, 0, 7, 2, 4, 6, 1, 8, 3, 5, 0, 7, 2, 4, 6, 1, 8, 3, 5, 0, 7, 2, 9, 4, 6, 1, 8, 3, 5, 0, 7, 2, 9, 4, 6, 1, 8, 3, 10, 5, 0, 7, 2, 9, 4, 6, 1, 8, 3, 10, 5, 0, 7

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 1000 terms from Franklin T. Adams-Watters)

Crossrefs

Cf. A003586, A022328, A191476. - Franklin T. Adams-Watters, Mar 19 2009

Cf. A069352.

Programs

Haskell

a022329 n = a022329_list !! (n-1)
-- Where a022329_list is defined in A022328.
-- Reinhard Zumkeller, Nov 19 2015, May 16 2015

Mathematica

IntegerExponent[Select[Range[10^5], # == 2^IntegerExponent[#, 2] * 3^IntegerExponent[#, 3] &], 3] (* Amiram Eldar, Apr 15 2024 *)

Python

from sympy import integer_log
def A022329(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1))
    return integer_log((m:=bisection(f,n,n))>>(~m&m-1).bit_length(),3)[0] # Chai Wah Wu, Sep 15 2024

Formula

a(n) = A069352(n) - A022328(n). - Reinhard Zumkeller, May 16 2015

A003586(n) = 2^A022328(n) * 3^a(n). - N. J. A. Sloane, Mar 19 2009

a(n) = A191476(n) - 1. - Franklin T. Adams-Watters, Mar 19 2009

Extensions

Edited by N. J. A. Sloane, May 26 2024