Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Path: utzoo!watmath!clyde!burl!ulysses!unc!mcnc!ncsu!uvacs!gmf
From: gmf@uvacs.UUCP
Newsgroups: net.math
Subject: Iterated sums of digits of divisors
Message-ID: <1297@uvacs.UUCP>
Date: Thu, 10-May-84 14:58:25 EDT
Article-I.D.: uvacs.1297
Posted: Thu May 10 14:58:25 1984
Date-Received: Sat, 12-May-84 12:55:22 EDT
Lines: 27


In the magazine * Abacus * for Winter, 1984 (v. 1, no. 2, Springer-Verlag)
there is the following problem on p. 73 (attributed to Dr. Herta Freitag
(Hollins College, retired):

Let N(0) be an integer > 1.  Define N(K+1) as the sum of the  * digits *
of the divisors of N(K)  [not the sum of the divisors!!] .  Prove or
disprove that all such sequences end up with N(K+1) = 15, which then
repeats.  Example with N(0) = 12:

     K     N(K)     Divisors of N(K)
----------------------------------------
     0     12       1 2 3 4 6 12
     1     19       1 19
     2     11       1 11
     3      3       1 3
     4      4       1 2 4
     5      7       1 7
     6      8       1 2 4 8
     7     15       1 3 5 15
     8     15  which repeats forever

I ran a couple of hundred on a machine and it worked every time (didn't
always arrive at 15 in the same way, but arrived).  Does anyone know why
this is?

     Gordon Fisher