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(PARI) A253074_upto(n=99, a, u, U)={vector(n, n, for(k=u, oo, bittest(U, k-u)|| isprime(a+k)||[a=k, break]); (a>u && U+=1<<(a-u))|| U>>=-u+u+=valuation(U+2, 2); a)+if(default(debug), print([u]))} \\ additional args allow to tweak computation. If debug > 0, print least unused number at the end. - M. F. Hasler, Nov 25 2019

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A simpler Simplified version of the proof: Assume x isn't in the sequence, then eventually all terms must be of the form PRIME - x, else x would appear next. In particular, no multiple of x can appear from there on. Assume k*x is the largest multiple of x in the sequence. Take a prime p not dividing x. Then m*x can't appear in the sequence for k+1 <= m <= k+p, and all terms are eventually of the form PRIME - m*x for all m in {k+1, ..., k+p}. Take one such term N > p, i.e., N + (k+1)*x, ..., N + (k+p)*x are all prime. Consider this sequence mod p. Since gcd(x,p)=1, the p terms cover each residue class mod p, so one is a multiple of p, in contradiction with their primality. - M. F. Hasler, Nov 25 2019

In particular, this means there are only finitely many multiples of x that appear in the sequence (because . Let Y be a multiple of x can't be larger than all multiples of x appearing in the form PRIME - x)sequence.

Let Y be a multiple of x larger than all multiples of x appearing in the sequence.

A simpler proof: Assume x isn't in the sequence, then eventually all terms must be of the form PRIME - x, else x would appear next. In particular, no multiple of x can appear from there on. Assume k*x is the largest multiple of x in the sequence. Take a prime p not dividing x. Then m*x can't appear in the sequence for k+1 <= m <= k+p, and all terms are eventually of the form PRIME - m*x for all m in {k+1, ..., k+p}. Take one such term N > p, i.e., N + (k+1)*x, ..., N + (k+p)*x are all prime. Consider this sequence mod p. Since gcd(x,p)=1, the p terms cover each residue class mod p, so one is a multiple of p, in contradiction with their primality. - M. F. Hasler, Nov 25 2019

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In particular, this means there are only finitely many multiples of x that appear in the sequence (because k*a multiple of x = can't be of the form PRIME - x <=> (k+1)*x = PRIME is impossible).

In particular, this means there are only finitely many multiples of x that appear in the sequence. To make this cleaner, let Y be a multiple of (because k*x = PRIME - x larger than all multiples of <=> (k+1)*x appearing in the sequence= PRIME is impossible).

Let Y be a multiple of x larger than all multiples of x appearing in the sequence.

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