

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
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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.  ZhiWei Sun, Jun 13 2015


REFERENCES

ZhiWei 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 ChinaJapan Seminar (Fukuoka, Oct. 28Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169187.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Checking the conjecture for r = n/m with 1 <= n <= m <= 500
ZhiWei 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

ZhiWei Sun, Jun 10 2015


STATUS

approved



