> I.e. Out of thirteen consecutive integers from the sequence > of the 30k+1, can we get at least 12 primes out > of the thirteen numbers, for the right choice > of the 13 consecutive numbers ?
Of those 13 consecutive numbers, one or two are divisible by 7; one or two are divisible by 11, one is divisible by 13, at most one divisible by 17 etc. To have only one divisible by 7, it must be the middle one. To have only one number composite, that number must also be divisible by 11 and 13. 1001 = 7x11x13. So you need to check