A191476 Values of j in the numbers 2^i*3^j, i >= 1, j >= 1, arranged in increasing order (A033845).

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

Offset: 1

Author

Zak Seidov, Aug 30 2012

Keywords

• nonn

Comments

This is the signature sequence of log(2)/log(3) (compare A022328). - N. J. A. Sloane, May 26 2024

Examples

			a(10) = 3 because A033845(10) = 108 = 2^2*3^3.
a(100) = 7 because A033845(100) = 59872 = 2^8*3^7.
a(1000) = 1 because A033845(1000) = 216172782113783808 = 2^56*3^1.
		

Links

• Zak Seidov, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to signature sequences

Crossrefs

Cf. A033845 (numbers 2^i*3^j), A191475 (values of i).

A022329 (= a(n)-1) is an essentially identical sequence.

See also A022328.

Programs

Mathematica

mx = 1000000; t = Select[Sort[Flatten[Table[2^i 3^j, {i, Log[2, mx]}, {j, Log[3, mx]}]]], # <= mx &]; Table[FactorInteger[i][[2, 2]], {i, t}] (* T. D. Noe, Aug 31 2012 *)

Python

from sympy import integer_log
def A191476(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 1+integer_log((m:=bisection(f,n,n))>>(~m&m-1).bit_length(),3)[0] # Chai Wah Wu, Sep 15 2024

Extensions

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