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If you string together all the prime factors of a number, and repeat, stopping when you reach a prime, you get the classic sequence of home primes, A037274 (where the 49th term is still unknown after all these years--see A056938).
Eric Angelini recently suggested stringing together all the proper divisors of the number, that is, all divisors except 1 and the number itself.
A120712 is the first of these new entries: it gives numbers n such that the concatenation of the proper divisors of n is a prime.
But look at A120716: start at n and repeatedly concatenate the proper divisors until you reach a prime, setting the value to -1 if we never reach a prime.
For example, the proper divisors of 6 are 2 and 3, so 6 -> 23, and since 23 is a prime, a(6) = 23. The big question is, what is a(8)? The beginning of the trajectory of 8 can be found here. There is a very large number that needs to be factored!
At lunch the other day we came up with four variants of A120716, hoping to find one where we could compute more terms: A130139, A130140, A130141, A130142.
But in every case we quickly get stopped. A pessimist might say that the human race will never find the next term in any of the last five sequences mentioned. It would be nice to know more! [Aug 02 2007]
Here is another pair of possible duplicates: Are A097344 and A097345 identical? It seems so, but I would like to have a formal proof. [Aug 02 2007]
Added Jan 27 2008: Maximilian F. Hasler has shown that these two sequences are in fact different. They first differ at the 59-th term.
Email: njasloane@gmail.com