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

1,1

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

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Run-Length Encoding.

Example

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}}.

Mathematica

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

Prog

(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<<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)))
print(list(islice(agen(), 52))) # Michael S. Branicky, Feb 25 2023

Crossrefs

Cf. A100714, A000040. Primes in A043570.

Cf. A019434.

Sequence in context: A172988 A020604 A268476 * A141246 A288445 A087759

Adjacent sequences: A082551 A082552 A082553 * A082555 A082556 A082557

Keyword

nonn,base

Author

Randy L. Ekl, May 03 2003

Status

approved