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

1,1

Comments

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

Example

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

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}]

Crossrefs

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

Sequence in context: A307873 A259138 A082710 * A065322 A166261 A089381

Adjacent sequences: A240601 A240602 A240603 * A240605 A240606 A240607

Keyword

nonn

Author

Zhi-Wei Sun, Apr 09 2014

Status

approved