Problem 1688. Prime Sequences: AP-k Minimum Final Value

Welcome to Fun with Primes. Today we will find the Minimum Final Value AP-k sequences for n_max=3:12 given the primorial and knowledge that the solution is of the form a + b * k# * n.

The AP-k of n sequence is n_max+1 primes of the form a + b * k# * n where n=0:n_max. The value of "a" is a prime and k# is the primorial.

The primorial k# is the product of all primes ≤ k, e.g. 10# = 2 · 3 · 5 · 7.

Input: (k, n_max)

Output: [a, b] for the equation Prime = a + b * k# * n, n=0:n_max; Prime(n_max) must be the optimum minimum.

Value Range Limits: [a<150,000 , b<8 ]

Example:

(13, 13) yields [31385539,14 ]; 31385539 + 14·13#·n (End Prime 36850999)

Commentary:

(13, 16) has a non-minimal end [17, 11387819007325752 ] to give Primes=17 + 11387819007325752·13#·n

The current June 2013 record for n is 25 via PrimeGrid: 43142746595714191 + 23681770·23#·n