A076689 Smallest k such that k*prime(n)# + 1 is prime where prime(n)# is the n-th primorial number A002110(n).

1, 1, 1, 1, 1, 4, 8, 11, 4, 11, 1, 4, 7, 6, 14, 3, 5, 2, 7, 3, 6, 20, 2, 9, 20, 2, 5, 7, 31, 2, 12, 13, 24, 7, 39, 21, 35, 24, 22, 3, 21, 8, 9, 13, 39, 21, 29, 10, 3, 62, 52, 21, 3, 36, 28, 15, 18, 33, 7, 46, 33, 20, 14, 22, 41, 7, 27, 39, 20, 4, 4, 5, 15, 27, 1, 44, 99, 9, 52, 2, 27, 12

Offset: 1

Author

Jason Earls, Nov 10 2002

Keywords

• nonn

Comments

From Pierre CAMI, Sep 12 2017: (Start)

Conjectures:

lim_{N->infinity} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} n) = 1/2;

a(n)/n is always < 4.

This is certified for the first 3100 primes a(n)*prime(n)#+1.

(End)

Links

• Pierre CAMI, Table of n, a(n) for n = 1..3100
• Pierre CAMI, PFGW Sript

Crossrefs

Cf. A002110, A014545 (n for which k=1), A073917 (the primes).

Programs

Mathematica

With[{P = FoldList[Times, Prime@ Range@ 120]}, Table[k = 1; While[CompositeQ[k P[[n]] + 1], k++]; k, {n, Length@ P}]] (* Michael De Vlieger, Sep 18 2017 *)

PARI

a(n) = my(k=1, pr = prod(i=1, n, prime(i))); while (! isprime(k*pr+1), k++); k; \\ Michel Marcus, Oct 09 2017