%I #38 Apr 12 2023 20:05:31

%S 5,9,11,13,17,19,23,25,27,29,33,35,39,47,49,51,55,57,59,61,65,67,71,

%T 79,95,97,99,103,111,113,115,119,121,123,125,129,131,135,143,159,191,

%U 193,195,199,207,223,225,227,231,239,241,243,247

%N Numbers whose base-2 representation has exactly 3 runs.

%C Numbers of the form 2^n - 2^m + 2^k - 1 for n > m > k > 0. - _Robert Israel_, Jan 11 2018

%C A000051 \ {2, 3} is a subsequence, since the base-2 representation of a number of the form 2^k+1 > 3 consists of a single 1, followed by a block of k-1 0's, followed by a last single 1. Also, A000215 \ {3} is another subsequence, since the base-2 representation of a Fermat number 2^(2^k)+1 > 3 consists of a single 1, followed by a block of 2^k-1 0's, followed by a last single 1. - _Bernard Schott_, Mar 09 2023

%C Numbers k such that A005811(k) = 3. - _Michel Marcus_, Mar 10 2023

%H Robert Israel, <a href="/A043570/b043570.txt">Table of n, a(n) for n = 1..10000</a>

%e 115 = 1110011_2, which is a block of three 1's, followed by a block of two 0's, followed by a block of two 1's, so 115 is a term.

%p seq(seq(seq(2^n-2^m+2^k-1, k=1..m-1),m=n-1..2,-1),n=2..10); # _Robert Israel_, Jan 11 2018

%o (Python)

%o from itertools import count, islice

%o def agen(): yield from ((1<<k)-(1<<j)+(1<<i)-1 for k in count(1) for j in range(k-1, 1, -1) for i in range(1, j))

%o print(list(islice(agen(), 53))) # _Michael S. Branicky_, Feb 25 2023

%Y Cf. A005811.

%Y Cf. A000051, A000215.

%Y Cf. A082554 (subsequence of primes).

%K nonn,base

%O 1,1

%A _Clark Kimberling_