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A272478
Primes with a prime number of binary digits, and with a prime number of 1's and a prime number of 0's.
1
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
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
KEYWORD
nonn,base
AUTHOR
Andres Cicuttin, May 01 2016
STATUS
approved