A272441 Primes with a prime number of binary digits.

2, 3, 5, 7, 17, 19, 23, 29, 31, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223

Offset: 1

Author

Andres Cicuttin, Apr 30 2016

Keywords

• nonn
• base
• easy

Examples

			7 is a term since its binary representation has 3 bits, a prime.
67 is a term since its binary representation has 7 bits, a prime.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A120533 (analogous in base 10).

Programs

Mathematica

Select[Table[j, {j, 1, 1200}], (PrimeQ[#] && PrimeQ[Length@IntegerDigits[#, 2]]) &]
Select[Prime[Range[200]],PrimeQ[Length[IntegerDigits[#,2]]]&] (* Harvey P. Dale, Jun 04 2019 *)

PARI

isok(n) = isprime(n) && isprime(#binary(n)); \\ Michel Marcus, Apr 30 2016

PARI

forprime(d=2,13, forprime(p=2^(d-1),2^d, print1(p", "))) \\ Charles R Greathouse IV, May 01 2016

Python

from itertools import islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    d = 3
    yield from [2, 3]
    while True:
        yield from (i for i in range(2**(d-1)+1, 2**d, 2) if isprime(i))
        d = nextprime(d)
print(list(islice(agen(), 50))) # Michael S. Branicky, Dec 27 2023

Python

from sympy import primepi, primerange
def A272441(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(min(x,(1<Chai Wah Wu, Feb 03 2025