Let A_n defined in the same way as A159559, but with initial term prime(n); a(n) is the smallest number m such that A_(n+1)(m) - A_n(m) > 6 or a(n)=0 if there is no such m.

A277118 is a subsequence. In contrast to A277118, the sequence contains some terms greater than 17. The first occurrence for 19 is n = 7061; the first occurrence for 21 is n = 214612 and the first occurrence for 23 is n = 30572.