Hot Sequences

Blog of Especially Interesting Recent Sequences In

The On-Line Encyclopedia of Integer Sequences

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Notes on some recent sequences of interest.
Last modified Jan 27 2008.
For more information about these sequences, look them up in The On-Line Encyclopedia of Integer Sequences.

1. 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]
2. Thanks to Andrew Plewe, a large number of duplicate sequences in the OEIS have been eliminated this year. 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.

Neil Sloane

Email: njas@research.att.com