Concatenating the proper divisors of a number
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]
