A307533 - OEIS

%I #82 Jul 29 2019 04:33:46

%S 13,19,31,37,43,53,61,67,73,83,89,97,109,113,127,131,139,151,157,173,

%T 181,199,211,223,233,251,257,263,277,293,307,317,331,337,349,353,367,

%U 373,379,389,401,409,421,439,443,449,457,467,479,487,491,499,503,509,541

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

%C (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?

%C 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?

%C 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.

%H Robert Israel, <a href="/A307533/b307533.txt">Table of n, a(n) for n = 1..10000</a>

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

%p filter:= proc(n) isprime(n) and nops(numtheory:-factorset(n+2))=2 end proc:

%p select(filter, [seq(i,i=3..1000,2)]); # _Robert Israel_, Jul 28 2019

%t Select[Range[500], PrimeQ[#] && PrimeNu[# + 2] == 2 &] (* _Amiram Eldar_, Apr 14 2019 *)

%o (PARI) isok(p) = isprime(p) && (omega(p+2) == 2); \\ _Michel Marcus_, May 02 2019

%K nonn,less

%O 1,1

%A _Paolo Galliani_, Apr 13 2019