A260197 Least prime p such that pi(p*n) = prime(q*n) for some prime q, where pi(x) denotes the number of primes not exceeding x.

5, 277, 29, 17, 43, 103, 53, 31, 1571, 3089, 37, 593, 881, 3023, 277, 9257, 47, 1949, 9137, 311, 17011, 1039, 53, 59, 2153, 15331, 3617, 631, 44867, 61, 17351, 661, 821, 2339, 683, 1201, 34759, 62687, 20327, 59369, 71, 883, 40189, 9187, 1879, 7669, 2767, 3931, 8867, 8081, 79, 12401, 139, 4787, 6367, 277, 2903, 23671, 32839, 3659

Offset: 1

Author

Zhi-Wei Sun, Jul 19 2015

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n > 0. Also, for any n > 0, there are primes p and q such that pi(p*n) = q*n.

Examples

			 a(1) = 5 since pi(5*1) = 3 = prime(2*1) with 2 and 5 both prime.
a(2) = 277 since pi(277*2) = 101 = prime(13*2) with 13 and 277 both prime.
a(10) = 3089 since pi(3089*10) = 3331 = prime(47*10) with 47 and 3089 both prime.
		

References

• Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..300
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A000720, A237578.

Programs

Mathematica

PQ[n_,p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]/n]
Do[k=0;Label[aa];k=k+1;If[PQ[n,PrimePi[Prime[k]*n]],Goto[bb],Goto[aa]];Label[bb];Print[n, " ", Prime[k]];Continue,{n,1,60}]