A258803 Least prime p such that n = (prime(q)-1)/(prime(p)-1) for some prime q.

2, 2, 5, 3, 2, 13, 11, 2, 23, 3, 11, 29, 19, 397, 2, 67, 131, 31, 5, 2, 5, 7, 1039, 5, 7, 67, 3, 787, 2, 13, 83, 149, 2, 89, 47, 43, 31, 809, 3, 5, 2, 307, 5, 61, 41, 5, 67, 19, 11, 1447, 101, 13, 881, 2, 37, 31, 331, 11, 1033, 3, 19, 839, 2, 61, 163, 59, 41, 1163, 3, 353, 67, 7, 313, 11, 7, 7, 101, 2, 71, 19, 7, 127, 409, 53, 149, 401, 283, 3, 2, 191, 43, 157, 163, 13, 2, 31, 89, 19, 5, 3

Offset: 1

Author

Zhi-Wei Sun, Jun 10 2015

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n > 0. Moreover, for any integers s and t with |s| = |t| = 1, each positive rational number r can be written as (prime(p) + s)/(prime(q) + t) with p and q both prime. - Sun

I have verified the conjecture for all those rational numbers r = a/b with a, b = 1, ..., 500. - Zhi-Wei Sun, Jun 13 2015

Examples

			a(1) = 2 since 1 = (prime(2) - 1)/(prime(2) - 1) with 2 prime.
a(2) = 2 since 2 = (prime(3) - 1)/(prime(2) - 1) with 2 and 3 both prime.
a(14) = 397 since 14 = (prime(4021) - 1)/(prime(397) - 1) = (38053 - 1)/(2719 - 1) with 379 and 4021 both prime.
a(23) = 1039 since 23 = (prime(17209) - 1)/(prime(1039) - 1) = (190579 - 1)/(8287 - 1) with 1039 and 17209 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..10000
• Zhi-Wei Sun, Checking the conjecture for r = n/m with 1 <= n <= m <= 500
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641.

Crossrefs

Cf. A000040.

Programs

Mathematica

PQ[n_]:=PrimeQ[n] && PrimeQ[PrimePi[n]];
Do[k = 0; Label[bb]; k = k + 1; If[PQ[n * (Prime[Prime[k]] - 1) + 1], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", Prime[k]]; Continue, {n, 1, 100}] (* Sun *)
pq[n_] := PrimeQ@n && PrimeQ@ PrimePi@ n; a[n_] := Block[{k = 1}, While[!pq[1 + n*(Prime@ Prime@ k - 1)], k++]; Prime@k]; Array[a, 100] (* Giovanni Resta, Jun 11 2015 *)