[seqfan] Do 20 and 105 eventually reach a prime in these sequences?

[Alonso Del Arte]

I was looking for a sequence for which it would make sense to use
DeleteCases in the Mathematica program. The only thing I could come up with
uses Cases and Except instead of DeleteCases.

If we take the factorization of an integer, say 18 = 2 * 3^2, strip out the
operators and concatenate the integers, then repeat the operation, most
integers quickly reach a prime. Indeed, 232 = 2^3 * 29, and 2329 = 17 * 137,
and 17137 is prime.

Because of the way Mathematica reports integers factorizations, it is
necessary to remove the exponents of 1 (e.g., 18 = 2^1 * 3^2) before the
concatenation step.

Do 20 and 105 reach a prime under this process? With 20 we get 20, 225,
3252, 223271, 297699, 399233, 715623, 3263907, 32347303, ... I've taken this
to forty steps, and 4339779194757514315803243245042123341102411963 is
composite. Similary with 105: 105, 357, 3717, 32759, 174147, 358049, 379677,
3196661, 13245897, ... and at forty steps,
7903796151050019316957414263672508157280737 is composite.

Al