A240604 Primes p with pi(p), pi(2*p), pi(3*p) and pi(4*p) all prime, where pi(x) denotes the number of primes not exceeding x.

10909, 67247, 185869, 408379, 511111, 1297061, 1730461, 1732333, 2135347, 2266079, 2316203, 2978917, 3477737, 4337257, 4495739, 4691849, 6108461, 6407971, 6591163, 7462589, 7909507, 8165039, 8298337, 8948509, 11144083, 11961373, 15019049, 16074059, 16732561, 19316263

Offset: 1

Author

Zhi-Wei Sun, Apr 09 2014

Keywords

• nonn

Comments

Conjecture: For any positive integer n, there are infinitely many primes p with pi(k*p) (k = 1,...,n) all prime.

Examples

			a(1) = 10909 with 10909, pi(10909) = 1327, pi(2*10909) = 2447, pi(3*10909) = 3511 and pi(4*10909) = 4547 all prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..2280
• Zhi-Wei Sun, Problems on combinatorial properties of primes, preprint, arXiv:1402.6641, 2014.

Crossrefs

Cf. A000040, A000720, A006450, A237578, A239428, A239430, A239617.

Programs

Mathematica

p[j_,k_]:=p[j,k]=PrimeQ[PrimePi[j*Prime[Prime[k]]]]
p[k_]:=p[k]=p[2,k]&&p[3,k]&&p[4,k]
m=0;Do[If[p[k],m=m+1;Print[m," ",Prime[Prime[k]]]],{k,1,95041}]