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

1,1

Links

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

Example

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.

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 *)

Prog

(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<<i)-1))-primepi((1<<i-1)-1) for i in primerange(2, x.bit_length()+1))
    return bisection(f, n, n) # Chai Wah Wu, Feb 03 2025

Crossrefs

Cf. A120533 (analogous in base 10).

Sequence in context: A360781 A042994 A129692 * A155471 A158015 A042995

Adjacent sequences: A272438 A272439 A272440 * A272442 A272443 A272444

Keyword

nonn,base,easy,changed

Author

Andres Cicuttin, Apr 30 2016

Status

approved