A020914 Number of digits in the base-2 representation of 3^n.

1, 2, 4, 5, 7, 8, 10, 12, 13, 15, 16, 18, 20, 21, 23, 24, 26, 27, 29, 31, 32, 34, 35, 37, 39, 40, 42, 43, 45, 46, 48, 50, 51, 53, 54, 56, 58, 59, 61, 62, 64, 65, 67, 69, 70, 72, 73, 75, 77, 78, 80, 81, 83, 85, 86, 88, 89, 91, 92, 94, 96, 97, 99, 100, 102, 104, 105, 107

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• base
• easy

Comments

Also, numbers k such that the first digit in the ternary expansion of 2^k is 1. - Mohammed Bouayoun (Mohammed.bouayoun(AT)sanef.com), Apr 24 2006

a(n) is the smallest integer such that n/a(n) < log_2(3). - Trevor G. Hyde (thyde12(AT)amherst.edu), Jul 31 2008

This sequence represents allowable values of the "dropping time" in the Collatz (3x+1) problem when iterated according to the function f(n) := n/2 if n is even, (3n+1)/2 otherwise, as tabulated in A126241. There is one exception, A126241(1), which has been set to zero by convention. - K. Spage, Oct 22 2009

An integer k is a term of A020914 if and only if floor(k*(1 + log(2)/log(3))) - abs(k-1)*(1 + log(2)/log(3)) - 1 >= 0. - K. Spage, Oct 22 2009

Also smallest k such that ceiling(2^k / 3^n) = 2. - Michel Lagneau, Jan 31 2012

For n > 0, first differences of A022330. - Michel Marcus, Oct 03 2013

Also the number of powers of two less than or equal to 3^n. - Robert G. Wilson v, May 25 2014

Except for 1, A020914 is the complement of A054414 and therefore these two form a pair of Beatty sequences. - Robert G. Wilson v, May 25 2014

Links

• T. D. Noe, R. J. Mathar, Table of n, a(n) for n = 0..20000
• Mike Winkler, On the structure and the behaviour of Collatz 3n + 1 sequences, 2014.
• Mike Winkler, New results on the stopping time behaviour of the Collatz 3x + 1 function, arXiv:1504.00212 [math.GM], 2015.
• Mike Winkler, The algorithmic structure of the finite stopping time behavior of the 3x + 1 function, arXiv:1709.03385 [math.GM], 2017.

Crossrefs

Cf. A056576, A054414, A070939, A000244, A227048, A022330, A022921 (first differences), A126241.

Cf. A020857 (decimal expansion of log_2(3)).

Cf. A020915.

Cf. A204399 (essentially the same).

Programs

Haskell

a020914 = a070939 . a000244  -- Reinhard Zumkeller, Jun 30 2013

Maple

A020914 :=n->nops(convert(3^n,base,2)):
seq(A020914(n),n=0..70); # Emeric Deutsch, Apr 30 2006
seq(ilog2(3^n)+1, n=0 .. 100); # Robert Israel, Dec 12 2014

Mathematica

Table[Length[IntegerDigits[3^n, 2]], {n, 0, 100}] (* Stefan Steinerberger, Apr 19 2006 *)
a[n_] := Floor[ Log2[3^n] + 1]; Array[a, 105, 0] (* Robert G. Wilson v, May 25 2014 *)

PARI

for(n=0,100,print1(floor(1+n*log(3)/log(2)),",")) \\ K. Spage, Oct 22 2009

PARI

a(n)=exponent(3^n)+1 \\ Charles R Greathouse IV, Nov 03 2022

Python

def A020914(n): return (3**n).bit_length() # Chai Wah Wu, Oct 09 2024

Formula

a(n) = floor(1 + n*log(3)/log(2)). - K. Spage, Oct 22 2009

a(0) = 1, a(n+1) = a(n) + A022921(n). - K. Spage, Oct 23 2009

a(n) = A122437(n-1) - n. - K. Spage, Oct 23 2009

A098294(n) = a(n) + n for n > 0. - Mike Winkler, Dec 31 2010

a(n) = A070939(A000244(n)) = length of n-th row in triangle A227048. - Reinhard Zumkeller, Jun 30 2013

a(n) = 1 + floor(n*log_2(3)) = 1 + A056576(n) = 1 + floor(n*A020857). - L. Edson Jeffery, Dec 12 2014

A020915(a(n)) = n + 1. - Reinhard Zumkeller, Mar 28 2015

Extensions

More terms from Stefan Steinerberger, Apr 19 2006