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

1,1

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

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10859

Michel Marcus, 3 primes

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A052294, A095079, A272441.

Sequence in context: A226681 A005808 A180559 * A028489 A210242 A165320

Adjacent sequences: A272475 A272476 A272477 * A272479 A272480 A272481

Keyword

nonn,base

Author

Andres Cicuttin, May 01 2016

Status

approved