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

1,1

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.

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.

Example

 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.

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}]

Crossrefs

Cf. A000040, A000720, A237578.

Sequence in context: A112901 A213958 A158115 * A225781 A368754 A057209

Adjacent sequences: A260194 A260195 A260196 * A260198 A260199 A260200

Keyword

nonn

Author

Zhi-Wei Sun, Jul 19 2015

Status

approved