A152609 a(1) = 1; thereafter a(n) is always the smallest integer > a(n-1) not leading to a contradiction, such that the concatenation of any four consecutive digits in the sequence is a prime.

1, 2, 3, 7, 19, 31, 91, 373, 931, 1931, 1933, 7193, 11931, 19311, 93119, 311931, 913733, 1373313, 7331373, 9311931, 19311931, 19311933, 71931193, 119311931, 193119311, 931193119, 3119311931, 9137331373, 9311931193