A333355 Number of bits in binary expansion of n minus the number of digits of n when written in base 3.

0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Author

Frank Ellermann, Mar 15 2020

Keywords

• nonn
• base
• easy

Comments

Record highs are at n = 2^A054414. All n=2^k >= 2 are increases, all n=3^j are decreases, and there is either one or none 3^j between 2^(k-1) and 2^k. When one, a(2^k) = a(2^(k-1)) so not a record high. When none, a(2^k) = a(2^(k-1)) + 1 which is a record high. If 2^k and 2^(k-1) are the same length in ternary then there is no 3^j between them. This is when 2^k has most significant ternary digit 2 since 2^(k-1) >= 3^j is 2^k >= 2*3^j. These k are A054414. Non-record increases are at its complement n = 2^A020914 >= 2. - Kevin Ryde, Apr 04 2020

Examples

			a(8) = 2 = 4 - 2 for binary 1000 and ternary 22.
a(64) = 3 = 7 - 4 for binary 1000000 and ternary 2101.
		

Crossrefs

Cf. A007088 ( binary), A000523 (floor(log_2(n))), A029837.

Cf. A007089 (ternary), A062153 (floor(log_3(n))), A117966.

Programs

Maple

a:= n-> ilog[2](n)-ilog[3](n):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 15 2020

Mathematica

a[n_]: = Floor @ Log[2, n] - Floor @ Log[3, n]; Array[a, 100] (* Amiram Eldar, Mar 16 2020 *)

PARI

a(n) = logint(n,2) - logint(n,3); \\ Kevin Ryde, May 15 2020

Rexx

L = 1 ;  M = 1 ;  B = 2 ;  T = 3       ;  S = 0
do N = 2 while length( S ) < 258
   if B = N then  do    ;  B = B * 2   ;  L = L + 1   ;  end
   if T = N then  do    ;  T = T * 3   ;  M = M + 1   ;  end
   S = S || ',' L - M
end N
say S                   ;  return S

Formula

a(n) = A000523(n) - A062153(n) = floor(log_2(n)) - floor(log_3(n)).

a(n) = length(A007088(n)) - length(A007089(n)).