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%I #21 Jun 15 2015 04:12:25
%S 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,
%T 3,787,2,13,83,149,2,89,47,43,31,809,3,5,2,307,5,61,41,5,67,19,11,
%U 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
%N Least prime p such that n = (prime(q)-1)/(prime(p)-1) for some prime q.
%C 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
%C I have verified the conjecture for all those rational numbers r = a/b with a, b = 1, ..., 500. - _Zhi-Wei Sun_, Jun 13 2015
%D 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.
%H Zhi-Wei Sun, <a href="/A258803/b258803.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="/A258803/a258803.txt">Checking the conjecture for r = n/m with 1 <= n <= m <= 500</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/14">Problems on combinatorial properties of primes</a>, arXiv:1402.6641.
%e a(1) = 2 since 1 = (prime(2) - 1)/(prime(2) - 1) with 2 prime.
%e a(2) = 2 since 2 = (prime(3) - 1)/(prime(2) - 1) with 2 and 3 both prime.
%e a(14) = 397 since 14 = (prime(4021) - 1)/(prime(397) - 1) = (38053 - 1)/(2719 - 1) with 379 and 4021 both prime.
%e a(23) = 1039 since 23 = (prime(17209) - 1)/(prime(1039) - 1) = (190579 - 1)/(8287 - 1) with 1039 and 17209 both prime.
%t PQ[n_]:=PrimeQ[n] && PrimeQ[PrimePi[n]];
%t 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 *)
%t 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 *)
%Y Cf. A000040.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Jun 10 2015
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