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

1,1

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.

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.

Example

a(4) = 67 since pi(67*4) = 56 = 4*14 = 4*pi(11*4) with 11 and 67 both prime.

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]<n*f[Prime[j]*n], Goto[bb]]; Continue, {j, 1, k}]; Label[aa]; Print[n, " ", Prime[k]]; Continue, {n, 1, 50}]

Crossrefs

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

Sequence in context: A306827 A028856 A013497 * A262366 A128029 A215167

Adjacent sequences: A260229 A260230 A260231 * A260233 A260234 A260235

Keyword

nonn

Author

Zhi-Wei Sun, Jul 20 2015

Status

approved