### Solution to Problem #2

A. Teitelman of Israel solved the problem for squares.
Ed Murphy of Granada Hills CA and
Dan Dima of Constanta (Romania) showed that the sequence would always end up in a loop
(regardless of the exponent).
Matthew Charlap of Columbus NJ,
Al Zimmermann of New York NY,
John T. Robinson of Yorktown Heights NY,
Matt Hudelson of Washington State University,
and
Philippe Fondanaiche of Paris (France)
completely characterized which loops occurred for squares and cubes.
Claudio Baiocchi of Gavignano (Italy) extended this to fourth powers
and Ross Millikan of San Mateo CA to fifth powers.
Jan van Delden of Zuidhorn (The Netherlands) extended the results
up to seventh powers.
Phillipe Fondanaiche and John T. Robinson note that there has been a lot of study in this area.
A number such that the iterates of the sum of the squares of its digits eventually ends in 1,
is called a *happy* number, see
en.wikipedia.org/wiki/Happy_number and
www.research.att.com/~njas/sequences/A007770.
The fates of some of the unhappy numbers are chronicled here:
www.research.att.com/~njas/sequences/A000216,
www.research.att.com/~njas/sequences/A000218,
www.research.att.com/~njas/sequences/A000221.
There is a MathWorld article on iterated digital power sums
here.

Jan van Delden's solution is here.

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