A060324 a(n) is the minimal prime q such that n*(q+1)-1 is prime, that is, the smallest prime q so that n = (p+1)/(q+1) with p prime; or a(n) = -1 if no such q exists.

2, 2, 3, 2, 3, 2, 5, 2, 5, 2, 3, 3, 7, 2, 3, 2, 3, 2, 5, 2, 3, 5, 5, 2, 5, 3, 3, 2, 5, 2, 13, 3, 3, 2, 3, 2, 11, 2, 5, 5, 3, 3, 5, 2, 3, 2, 5, 3, 5, 2, 19, 5, 3, 7, 7, 2, 3, 2, 5, 2, 7, 11, 3, 2, 5, 2, 5, 3, 11, 5, 3, 5, 13, 5, 5, 2, 3, 2, 7, 2, 7, 5, 3, 2, 5, 2, 3, 2, 17, 2, 7, 3, 5, 2, 3, 3, 11, 2, 5, 5

Offset: 1

Author

Matthew Conroy, Mar 29 2001

Keywords

• nonn
• nice
• easy

Comments

A conjecture of Schinzel, if true, would imply that such a q always exists.

Examples

			1 = (2+1)/(2+1), so the first term is 2; 3(2+1) - 1 = 8 which is not prime, yet 3(3+1) - 1 = 11 is prime (3 = (11+1)/(3+1)) so the 3rd term is 3.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Matthew M. Conroy, A sequence related to a conjecture of Schinzel, J. Integ. Seqs. Vol. 4 (2001), #01.1.7.
• Peter Luschny, Schinzel-Sierpinski conjecture and Calkin-Wilf tree.

Crossrefs

Cf. A060424. Values of p are given in A062251.

Cf. A000040, A010051.

Programs

Haskell

a060324 n = head [q | q <- a000040_list, a010051' (n * (q + 1) - 1) == 1]
-- Reinhard Zumkeller, Aug 28 2014

Maple

a:= proc(n) local q;
       q:= 2;
       while not isprime(n*(q+1)-1) do
          q:= nextprime(q);
       od; q
    end:
seq(a(n), n=1..300); # Alois P. Heinz, Feb 11 2011

Mathematica

a[n_] := (q = 2; While[!PrimeQ[n*(q + 1) - 1], q = NextPrime[q]]; q); a /@ Range[100] (* Jean-François Alcover, Jul 20 2011, after Maple prog. *)

PARI

a(n) = {my(q=2); while (!isprime(n*(q+1)-1), q = nextprime(q+1)); q;} \\ Michel Marcus, Nov 20 2017

Formula

a(n) = (A062251(n)+1) / n - 1. - Reinhard Zumkeller, Aug 28 2014