## Pseudo-Fortunate Numbers

### Problem 293

Published on 22 May 2010 at 05:00 am [Server Time]

An even positive integer N will be called admissible, if it is a power of 2 or its distinct prime factors are consecutive primes.

The first twelve admissible numbers are 2,4,6,8,12,16,18,24,30,32,36,48.

If N is admissible, the smallest integer M 1 such that N+M is prime, will be called the pseudo-Fortunate number for N.

For example, N=630 is admissible since it is even and its distinct prime factors are the consecutive primes 2,3,5 and 7.

The next prime number after 631 is 641; hence, the pseudo-Fortunate number for 630 is M=11.

It can also be seen that the pseudo-Fortunate number for 16 is 3.

Find the sum of all distinct pseudo-Fortunate numbers for admissible numbers N less than 10^{9}.

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