A303436 Primes p such that all the composite numbers between p and its next prime have no more than 2 distinct prime factors.

2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 43, 47, 53, 71, 79, 97, 107, 157, 191, 223, 431, 499, 673, 1151, 1213, 2591, 51199, 139967, 472391, 703123, 786431, 995327, 57395627, 63700991, 169869311, 4076863487, 10871635967

Offset: 1

Author

Amiram Eldar, Apr 24 2018

Keywords

• nonn
• more

Comments

Supersequence of A078883. Terms that are not there: 2, 7, 13, 19, 23, 31, 37, 43, 47, 53, 79, 97, 157, 223, 499, 673, 1213, 51199, 703123, ...

5*10^11 < a(39) <= 2348273369087. - Giovanni Resta, Apr 26 2018

Examples

			157 is in the sequence since it is a prime, and the composite numbers between it and its next prime, 163, have only 2 distinct prime factors: 158 = 2*79, 159 = 3*53, 160 = 2^5*5, 161 = 7*23, and 162 = 2*3^4.
		

Crossrefs

Cf. A078883.

Programs

Mathematica

b[n_] := Max[Map[PrimeNu, Range[n + 1, NextPrime[n] - 1]]]; c[n_] := b[Prime[n]]; a={}; Do[If[c[n] < 3, AppendTo[a, Prime[n]]], {n, 1, 10^7}]; a

PARI

isok(p) = {if (isprime(p), for(c=p+1, nextprime(p+1)-1, if (omega(c) != 2, return(0));); return (1););} \\ Michel Marcus, Apr 26 2018

Extensions

a(36) from Michel Marcus, Apr 26 2018

a(37)-a(38) from Giovanni Resta, Apr 26 2018