Newsgroups: rec.puzzles
From: hoey@zogwarg.etl.army.mil (Dan Hoey)
Date: 8 Jan 91 15:08:11 GMT
Subject: An old Math Magazine problem (was Re: A mysterious shell script.)

s...@Lise.Unit.NO (Stig Hemmer) writes:

>So it does. The earlier post asked the following question: (paraphrased)
>'Given a start number X_0, make the sequence X_1=f(X_0), X_2=f(X_1) etc.
> Will two such sequences (from different starting numbers) always merge?'

Where f(x) was described as the sum of all the factors of x, including
1 and x.  This function is usually called sigma(x).

Unfortunately, Stig misparaphrases the previous article, in which
amb...@acf5.NYU.EDU ([garbled] Balamurali Ambati) asked

>This is an old Mathematics Magazine Problem from ~ 15 years ago.  In
>a sequence of positive integers, N_{k+1} = N_k + the sum of all the
>distinct prime factors of N_k including 1 and N_k, k=0,1,2,...

Or, if we standardize the nomenclature, N_k plus the sum of the
distinct prime factors of N_k, including N_k if prime, plus the
non-prime 1.

Unfortunately, Ambati continues

>Such a sequence is 1,2,3,4,7,8,11,12,18,24,...  The sequence
>5,6,12,18,...  merges with the first sequence as do all sequences
>with N_0 < 91.

In which it is clear that N_{k+1} is taken to be N_k plus the sum of
the distinct prime factors of N_k, excluding N_k if prime, plus the
non-prime 1.

I do not know which sequence appeared in the Math Magazine problem.
If anyone looks it up, please let me know.

Dan Hoey
Hoey@Zogwarg.ETL.Army.Mil