A247068 Primes whose base-2 expansion has no two consecutive 1's.

2, 5, 17, 37, 41, 73, 137, 149, 257, 277, 293, 337, 521, 577, 593, 641, 661, 673, 677, 1033, 1061, 1093, 1097, 1109, 1153, 1193, 1289, 1297, 1301, 1321, 1361, 2053, 2069, 2081, 2089, 2113, 2129, 2213, 2309, 2341, 2377, 2389, 2593, 2633, 2689, 2693, 2729, 4129, 4133, 4177, 4229, 4241, 4261, 4357, 4373, 4421, 4649, 4673, 5153, 5189

Offset: 1

Author

Jeffrey Shallit, Nov 16 2014

Keywords

• nonn
• base

Comments

Also: numbers appearing in both A000040 and A003714. Is it known to be infinite?

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Estelle Basor, Brian Conrey, and Kent E. Morrison, Knots and ones, arXiv:1703.00990 [math.GT], 2017. See page 1.

Crossrefs

Cf. A000040, A003714.

Programs

Maple

M:= 16: # to get all terms < 2^M
B1:= {1}:
B2:= {}:
for n from 2 to M-1 do
   B3:= map(`+`,B1,2^n);
   B1:= B1 union B2;
   B2:= B3;
od:
select(isprime,{2} union B1 union B2);
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%,list));   # Robert Israel, Nov 16 2014

Mathematica

Select[Prime[Range[700]],SequenceCount[IntegerDigits[#,2],{1,1}]==0&] (* Harvey P. Dale, May 14 2022 *)

PARI

my(t=bitand(n++,2*n));if(t==0,return(n));my(o=#binary(t)-1);((n>>o)+1)<Charles R Greathouse IV, Nov 16 2014

PARI

forprime(p=2,5000,if(bitand(p,p>>1)==0,print1(p,", "))); \\ Joerg Arndt, Apr 25 2025

Python

from itertools import islice
from sympy import isprime
def A247068_gen(): # generator of terms
    k = 0
    while True:
        if isprime(k:=(m:=~(k>>1))&(k-m)): yield k
A247068_list = list(islice(A247068_gen(),30)) # Chai Wah Wu, Apr 25 2025

Sage

def a_list(M):  # All terms < 2^M. After Robert Israel.
    A = [1]; B = [2]; s = 4
    for n in range(M-2):
        C = [a + s for a in A]
        A.extend(B)
        B = C
        s <<= 1
    A.extend(B)
    return list(filter(is_prime, A))
a_list(13) # Peter Luschny, Nov 16 2014