A231018 a(n) = d(n)/p(n-1)# where d(n) > 0 is the common difference of the smallest p-term arithmetic progression of primes beginning with p = p(n) = n-th prime.

1, 1, 1, 5, 7315048, 4293861989, 11387819007325752, 4244193265542951705

Offset: 1

Author

Jonathan Sondow, Nov 08 2013

Keywords

• nonn
• hard
• more

Comments

d(n) is the least d > 0 such that p, p+d, p+2d, ..., p+(p-1)d are all prime with p = p(n), and p(n-1)# = A002110(n-1) is a primorial.

d(n) is always a multiple of p(n-1)#.

a(5) and a(6) are due to G. Loh in 1986, and a(7) to Phil Carmody in 2001.

See A088430 and A231017 for more comments, references, links, and examples.

Examples

			Prime(3) = 5 and 5, 11, 17, 23, 29 is the smallest 5-term AP beginning with 5, so a(3) = (11-5)/p(2)# = 6/2*3 = 1.
		

Links

• Index entries for sequences related to primes in arithmetic progressions

Crossrefs

Cf. A002110, A088430, A231017.

Formula

a(n) = A088430(n) / A002110(n) = (A231017(n) - prime(n)) / A002110(n).

Extensions

a(8) due to Wojciech Izykowski in 2014 added by Jonathan Sondow, Aug 08 2019