A307533 Primes p such that p+2 has exactly two distinct prime factors.

13, 19, 31, 37, 43, 53, 61, 67, 73, 83, 89, 97, 109, 113, 127, 131, 139, 151, 157, 173, 181, 199, 211, 223, 233, 251, 257, 263, 277, 293, 307, 317, 331, 337, 349, 353, 367, 373, 379, 389, 401, 409, 421, 439, 443, 449, 457, 467, 479, 487, 491, 499, 503, 509, 541

Offset: 1

Author

Paolo Galliani, Apr 13 2019

Keywords

• nonn
• less

Comments

(13,31), (37,73), (157,751), (199,991) are pairs of emirps belonging to this sequence such that the lesser term of the pair is the reverse of the greater. Are there infinitely many such pairs?

Are there infinitely many triples in the sequence like (61,67,73) and (251,257,263), that is, infinitely many a(n) such that a(n+1)=a(n)+6 and a(n+2)=a(n)+12?

The triples found so far are (61,67,73), (251,257,263) and (367,373,379). The first terms of the triples found are 61, 251 and 367, which belong to the sequence A038107.

Examples

			61 is in the sequence because 61 + 2 = 63 has exactly two distinct prime factors (3 and 7).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Programs

Maple

filter:= proc(n) isprime(n) and nops(numtheory:-factorset(n+2))=2 end proc:
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, Jul 28 2019

Mathematica

Select[Range[500], PrimeQ[#] && PrimeNu[# + 2] == 2 &] (* Amiram Eldar, Apr 14 2019 *)

PARI

isok(p) = isprime(p) && (omega(p+2) == 2); \\ Michel Marcus, May 02 2019