Terms were found by generating in ascending order the 13-smooth numbers up to some limit and checking the differences. The first 7 candidates k have been proved true by showing a set of 31 congruences to be impossible (see link below). For a discussion iof this method of proof see A308247.
35803 = A308247(6) cannot be written as the difference of 13-smooth numbers. All smaller numbers can; for example, 9007 = 3^2*7*11*13 - 2, 10091 = 2^2*3*5*13^2 - 7^2.
