A237659 Primes p with pi(p) and pi(p^2) both prime, where pi(.) is given by A000720.

17, 41, 59, 109, 127, 157, 353, 367, 709, 1153, 1787, 3319, 3407, 3911, 5851, 6037, 6217, 6469, 8389, 9103, 9319, 10663, 13709, 14107, 14591, 15683, 18433, 19463, 19577, 20107, 21727, 23209, 27809, 29383, 32797, 35023, 36251, 36599, 38351, 39239

Offset: 1

Author

Zhi-Wei Sun, Feb 11 2014

Keywords

• nonn

Comments

This is a subsequence of A237658.

Conjecture: The sequence has infinitely many terms.

Examples

			a(1) = 17 with pi(17) = 7 and pi(17^2) = 61 both prime.
a(2) = 41 with pi(41) = 13 and pi(41^2) = 263 both prime.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..600 from Zhi-Wei Sun)

Crossrefs

Cf. A000040, A000290, A000720, A006450, A038107, A237656, A237657, A237658.

Programs

Mathematica

p[m_]:=PrimeQ[PrimePi[m^2]]
n=0;Do[If[p[Prime[Prime[k]]],n=n+1;Print[n," ",Prime[Prime[k]]]],{k,1,1000}]
Select[Prime[Range[4500]],AllTrue[{PrimePi[#],PrimePi[#^2]},PrimeQ]&] (* Harvey P. Dale, May 10 2025 *)