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

1,1

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, Kent E. Morrison, Knots and ones, arXiv:1703.00990 [math.GT], 2017. See page 1.

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 *)

Prog

(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
(PARI) my(t=bitand(n++, 2*n)); if(t==0, return(n)); my(o=#binary(t)-1); ((n>>o)+1)<<o
n=0; while(n<1e6, if(isprime(n=step(n)), print1(n", "))) \\ Charles R Greathouse IV, Nov 16 2014

Crossrefs

Cf. A000040, A003714.

Sequence in context: A125822 A025537 A245784 * A028916 A100272 A107630

Adjacent sequences: A247065 A247066 A247067 * A247069 A247070 A247071

Keyword

nonn,base

Author

Jeffrey Shallit, Nov 16 2014

Status

approved