Take any odd number k, with distinct odd prime divisors p_1,...p_k and ask the following question: What is the smallest even number which when added to each prime divisor p_i of k, gives another prime? Example: k= 119=7*17, so the smallest even number is 6 because 7+6=11 and 17+6=23. NB: some numbers (eg 105,195,231...) seem to have no solution. I would appreciate assistance with a code to calculate for every odd k, the even number in question, or to allocate 0 for numbers where no solution has been found up to some suitably convincing high number like N= 10^5 or 10^6, (a parameter I could change if desired). The idea would be to conjecture that the apparently no solution numbers (up to N) don’t actually have a solution. Any assistance much appreciated in advance. I have found the solutions up to k=781 by hand, but my hand is getting tired now.
ps: I would like to have the option (if possible) to output the numbers with solution = 0, and the numbers which are the smallest to have solution 2*n, as separate sequences, for n up to some arbitrary value (ie 3,7,23,69,93...). (n>=1). Hope this is not asking too much.