A272478 Primes with a prime number of binary digits, and with a prime number of 1's and a prime number of 0's.

17, 19, 79, 103, 107, 109, 5119, 6079, 6911, 7039, 7103, 7151, 7159, 7919, 7927, 7933, 8059, 8111, 8123, 8167, 8171, 8179, 442367, 458239, 458719, 458747, 487423, 491503, 499711, 507839, 507901, 515839, 516091, 520063, 523007, 523261, 523519, 523759, 523771, 523903, 524219, 524221, 524269

Offset: 1

Author

Andres Cicuttin, May 01 2016

Keywords

• nonn
• look
• base

Comments

If the sum of primes p and q is a prime r, then one of p and q must be 2. - N. J. A. Sloane, May 01 2016

Examples

			a(3) = 79, its binary representation is 1001111 with (prime) 7 digits, (prime) 5 1's and (prime) 2 0's.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10859
• Michel Marcus, 3 primes

Crossrefs

Cf. A052294, A095079, A272441.

Cf. A006512 (bitlengths of terms).

Programs

Mathematica

Select[Table[Prime[j],{j,1,120000}],PrimeQ[Total@IntegerDigits[#,2]]&&PrimeQ[Length@IntegerDigits[#,2]]&&PrimeQ[(Length@IntegerDigits[#,2]-Total@IntegerDigits[#,2])]&]
Select[Prime@ Range[10^5], And[PrimeQ@ Total@ #, PrimeQ@ First@ #, PrimeQ@ Last@ #] &@ DigitCount[#, 2] &] (* Michael De Vlieger, May 01 2016 *)

PARI

isok(n) = isprime(n) && isprime(#binary(n)) && isprime(hammingweight(n)) && isprime(#binary(n) - hammingweight(n)); \\ Michel Marcus, May 01 2016

Python

from sympy import isprime, nextprime
from itertools import combinations, islice
def agen(): # generator of terms
    p = 2
    while True:
        p = nextprime(p)
        if not isprime(p+2): continue
        if isprime(t:=(1<<(p+1))+1): yield t
        b = (1<<(p+2))-1
        for i, j in combinations(range(p), 2):
            if isprime(t:=b-(1<<(p-i))-(1<<(p-j))):
                yield t
print(list(islice(agen(), 43))) # Michael S. Branicky, Dec 27 2023