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

%I #22 Dec 27 2023 11:58:06

%S 17,19,79,103,107,109,5119,6079,6911,7039,7103,7151,7159,7919,7927,

%T 7933,8059,8111,8123,8167,8171,8179,442367,458239,458719,458747,

%U 487423,491503,499711,507839,507901,515839,516091,520063,523007,523261,523519,523759,523771,523903,524219,524221,524269

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

%C 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

%H Alois P. Heinz, <a href="/A272478/b272478.txt">Table of n, a(n) for n = 1..10859</a>

%H Michel Marcus, <a href="/A272478/a272478.txt">3 primes</a>

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

%t Select[Table[Prime[j],{j,1,120000}],PrimeQ[Total@IntegerDigits[#,2]]&&PrimeQ[Length@IntegerDigits[#,2]]&&PrimeQ[(Length@IntegerDigits[#,2]-Total@IntegerDigits[#,2])]&]

%t Select[Prime@ Range[10^5], And[PrimeQ@ Total@ #, PrimeQ@ First@ #, PrimeQ@ Last@ #] &@ DigitCount[#, 2] &] (* _Michael De Vlieger_, May 01 2016 *)

%o (PARI) isok(n) = isprime(n) && isprime(#binary(n)) && isprime(hammingweight(n)) && isprime(#binary(n) - hammingweight(n)); \\ _Michel Marcus_, May 01 2016

%o (Python)

%o from sympy import isprime, nextprime

%o from itertools import combinations, islice

%o def agen(): # generator of terms

%o p = 2

%o while True:

%o p = nextprime(p)

%o if not isprime(p+2): continue

%o if isprime(t:=(1<<(p+1))+1): yield t

%o b = (1<<(p+2))-1

%o for i, j in combinations(range(p), 2):

%o if isprime(t:=b-(1<<(p-i))-(1<<(p-j))):

%o yield t

%o print(list(islice(agen(), 43))) # _Michael S. Branicky_, Dec 27 2023

%Y Cf. A052294, A095079, A272441.

%K nonn,base

%O 1,1

%A _Andres Cicuttin_, May 01 2016