cp's OEIS Frontend

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

%I A333355 #36 Dec 20 2024 19:12:28
%S A333355 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,
%T A333355 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,
%U A333355 3,3,3,3,3,3,3,3,3,3,3,3,2,2,2,2,2,2,2
%N A333355 Number of bits in binary expansion of n minus the number of digits of n when written in base 3.
%C A333355 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
%F A333355 a(n) = A000523(n) - A062153(n) = floor(log_2(n)) - floor(log_3(n)).
%F A333355 a(n) = length(A007088(n)) - length(A007089(n)).
%e A333355 a(8) = 2 = 4 - 2 for binary 1000 and ternary 22.
%e A333355 a(64) = 3 = 7 - 4 for binary 1000000 and ternary 2101.
%p A333355 a:= n-> ilog[2](n)-ilog[3](n):
%p A333355 seq(a(n), n=1..100);  # _Alois P. Heinz_, Mar 15 2020
%t A333355 a[n_]: = Floor @ Log[2, n] - Floor @ Log[3, n]; Array[a, 100] (* _Amiram Eldar_, Mar 16 2020 *)
%o A333355 (Rexx)
%o A333355 L = 1 ;  M = 1 ;  B = 2 ;  T = 3       ;  S = 0
%o A333355 do N = 2 while length( S ) < 258
%o A333355    if B = N then  do    ;  B = B * 2   ;  L = L + 1   ;  end
%o A333355    if T = N then  do    ;  T = T * 3   ;  M = M + 1   ;  end
%o A333355    S = S || ',' L - M
%o A333355 end N
%o A333355 say S                   ;  return S
%o A333355 (PARI) a(n) = logint(n,2) - logint(n,3); \\ _Kevin Ryde_, May 15 2020
%Y A333355 Cf. A007088 ( binary), A000523 (floor(log_2(n))), A029837.
%Y A333355 Cf. A007089 (ternary), A062153 (floor(log_3(n))), A117966.
%K A333355 nonn,base,easy
%O A333355 1,8
%A A333355 _Frank Ellermann_, Mar 15 2020