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

1,3

Comments

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

Links

Zak Seidov, Table of n, a(n) for n = 1..10000

Index entries for sequences related to signature sequences

Example

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.

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 *)

Prog

(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

Crossrefs

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

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

See also A022328.

Sequence in context: A292224 A023130 A084532 * A134583 A087467 A231568

Adjacent sequences: A191473 A191474 A191475 * A191477 A191478 A191479

Keyword

nonn

Author

Zak Seidov, Aug 30 2012

Extensions

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

Status

approved