A082554 Primes whose base-2 representation is a block of 1's, followed by a block of 0's, followed by a block of 1's.

5, 11, 13, 17, 19, 23, 29, 47, 59, 61, 67, 71, 79, 97, 103, 113, 131, 191, 193, 199, 223, 227, 239, 241, 251, 257, 263, 271, 383, 449, 463, 479, 487, 499, 503, 509, 769, 911, 967, 991, 1009, 1019, 1021, 1031, 1039, 1087, 1151, 1279, 1543, 1567, 1663, 1823

Offset: 1

Author

Randy L. Ekl, May 03 2003

Keywords

• nonn
• base

Comments

The n-th prime is a term iff A100714(n) = 3. - Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 11 2004

A019434 \{3} is a subsequence, since the base-2 representation of a Fermat prime 2^(2^k)+1 > 3 is a single 1, followed by a block of 2^k-1 0's, followed by a last single 1. - Bernard Schott, Mar 07 2023

Examples

			1987 = 11111000011_2, which is a block of 5 1's, followed by a block of 4 0's, followed by a block of 2 1's, so 1987 is a term.
a(3)=17 is a term because it is the 3rd prime whose binary representation splits into exactly three runs: 17_10 = 10001_2 splits into {{1}, {0,0,0}, {1}}.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Run-Length Encoding.

Crossrefs

Cf. A100714, A000040. Primes in A043570.

Cf. A019434.

Programs

Mathematica

Select[Table[Prime[k], {k, 1, 500}], Length[Split[IntegerDigits[ #, 2]]] == 3 &]

PARI

decomp(s)=if(s%2==0,return(1),); k=1; while(k==1,k=s%2; s=floor(s/2)); if(s==0,return(1),); while(k==0,k=s%2; s=floor(s/2)); while(k==1,k=s%2; s=floor(s/2)); return(s)
forprime(i=1,2000,if(decomp(i)==0,print1(i,", ")))

Python

from sympy import isprime
from itertools import count, islice
def agen(): yield from filter(isprime, ((1<Michael S. Branicky, Feb 25 2023