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

1,8

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

Links

Table of n, a(n) for n=1..87.

Formula

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

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

Example

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

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 *)

Prog

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

Crossrefs

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

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

Sequence in context: A030615 A336766 A147753 * A354523 A116531 A375039

Adjacent sequences: A333352 A333353 A333354 * A333356 A333357 A333358

Keyword

nonn,base,easy

Author

Frank Ellermann, Mar 15 2020

Status

approved