login
A258803
Least prime p such that n = (prime(q)-1)/(prime(p)-1) for some prime q.
10
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
1,1
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
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.
EXAMPLE
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.
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 *)
CROSSREFS
Cf. A000040.
Sequence in context: A361119 A293833 A146316 * A157495 A308143 A308515
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jun 10 2015
STATUS
approved