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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 (list; graph; refs; listen; history; text; internal format)
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.

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.

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: A132851 A293833 A146316 * A157495 A308143 A308515

Adjacent sequences:  A258800 A258801 A258802 * A258804 A258805 A258806

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jun 10 2015

STATUS

approved

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Last modified October 5 16:55 EDT 2022. Contains 357259 sequences. (Running on oeis4.)