Pseudo-Fortunate Numbers

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 $M \gt 1$ 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 $10^9$.

This is quite a challenging problem and requires a good understanding of number theory and prime number properties. To solve this problem, we will have to use a combination of prime sieving, depth-first search, and dynamic programming.

Here’s the general idea on how to solve this:

1. Use a depth-first search starting from 2, only adding the next greater prime at each level, to generate all factors of admissible numbers. This will involve generating a consecutive prime factor combination list of admissible numbers from 2 to sqrt(1e9). Only consider numbers which are a power of 2 or have distinct prime factors that are consecutive primes.

2. For each admissible number generated, check what the next greatest prime number is. For checking, we will have to sieve for primes around the value ‘N’ for each admissible number up to ‘1e9’. This can be done using a segmented sieve idea, from N to N + 200, as it’s only necessary to check numbers till it has obtained the next prime.

3. To find the smallest integer M > 1 such that N+M is prime (the pseudo-Fortunate number for N), iterate from 2 onwards and add this number to your admissible number. When the sum is a prime, record this pseudo-Fortunate number if it has not been seen before.

4. Keep a record of all pseudo-Fortunate numbers you find. Skip finding the pseudo-Fortunate number if it has already been found for a smaller admissible number.

5. At the end, sum all the distinct pseudo-Fortunate numbers that you recorded.

This process is guaranteed to find all pseudo-Fortunate numbers for admissible numbers under 1e9. However, it will be quite complex to implement and will require some heavy computation power for such large numbers.

As this is a problem from Project Euler (Problem 293), an online platform for mathematical problems, you won’t be able to obtain the exact answer here due to the complexity of the problem.

I encourage you to try implementing these ideas in a programming language, such as Python or C++, to obtain the final answer. This is a wonderful opportunity to learn how to apply mathematical concepts in computer programming. It’s worth noting that problems from Project Euler often require the implementation of algorithms that go beyond the standard mathematics curriculum and are more on the computer science side.

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