A160433 a(n) is the least number k such that (k-th prime after n!+1)-n! is not a prime.

2, 2, 3, 7, 8, 15, 8, 18, 16, 19, 12, 20, 11, 8, 11, 6, 12, 23, 24, 15, 31, 21, 27, 15, 16, 26, 25, 17, 17, 29, 20, 27, 27, 30, 23, 16, 28, 23, 25, 29, 15, 24, 19, 36, 36, 39, 15, 36, 24, 44, 35, 29, 27, 25, 36, 22, 37, 31, 32, 41, 29, 55, 27, 45, 29, 59, 34, 37, 24, 49, 25, 40

Offset: 0

Author

Frederick Magata (frederick.magata(AT)web.de), May 13 2009

Keywords

• nonn

Comments

The conjectures from A037153 and A087202 can be rephrased using a(n):

Is a(n)>=2 for all n>=0 and a(n)>=3 for all n>=2?

Also compare this with the conjecture on the fortunate numbers A005235.

Is the following true: for every m there is an N such that for all n>N a(n)>m?

There even seems to be the estimate a(n)>log(n+1)*sqrt(n+1)/2.

Examples

			a(3)=7: The seven primes following 3!+1=7 are 11,13,17,19,23,29 and 31.
Subtracting 3!=6 from each of them gives 5,7,11,13,17,23 and 25.
The first six values are prime, while the seventh 25=5^2 is not.
		

Crossrefs

Cf. A005235, A037153, A087202.

Programs

Maple

a:=proc(n) option remember; local k:
for k from 1 while isprime((nextprime@@k)(n!+1)-n!) do od:
k; end;