A260232 Least prime p such that pi(p*n) = n*pi(q*n) for some prime q.

2, 5, 13, 67, 23, 19, 433, 443, 107, 41, 61, 251, 239, 1987, 541, 491, 1093, 499, 421, 179, 2137, 1297, 1097, 101, 103, 2411, 1283, 1847, 379, 4993, 8329, 5563, 4297, 5639, 9587, 1867, 5113, 6691, 3691, 1193, 4663, 2971, 27733, 7121, 593, 2273, 607, 6047, 4217, 2609

Offset: 1

Author

Zhi-Wei Sun, Jul 20 2015

Keywords

• nonn

Comments

Conjecture: For any positive integer n, each rational number r > 0 can be written as pi(p*n)/pi(q*n) with p and q both prime.

For example, 4/7 = 416/728 = pi(479*6)/pi(919*6) with 479 and 919 both prime.

The conjecture holds trivially for n = 1 since pi(prime(m)*1) = m for all m = 1,2,3,.... Also, the conjecture implies that a(n) exists for any n > 0.

Examples

			a(4) = 67 since pi(67*4) = 56 = 4*14 = 4*pi(11*4) with 11 and 67 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, Checking the conjecture for n = 1..10 and r = a/b with a,b = 1..100
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A000720, A257364, A257928, A260140, A260252.

Programs

Mathematica

f[n_]:=PrimePi[n]; Do[k=0;Label[bb];k=k+1;If[Mod[f[Prime[k]*n],n]>0,Goto[bb]];Do[If[f[Prime[k]n]==n*f[Prime[j]*n],Goto[aa]];If[f[Prime[k]n]