A085416 Take prime[n] and continue adding 1,2,..., A085415(n) until one reaches a prime a(n).

3, 13, 11, 13, 17, 19, 23, 29, 29, 107, 37, 43, 47, 53, 53, 59, 137, 67, 73, 107, 79, 89, 89, 167, 103, 107, 109, 113, 137, 149, 137, 137, 173, 149, 227, 157, 163, 173, 173, 179, 257, 191, 197, 199, 233, 227, 239, 229, 233, 239, 239, 317, 251, 257, 263, 269, 347