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**
Smallest Examples of Strings of Consecutive Happy Numbers
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Robert Styer

Department of Mathematical Sciences

Villanova University

Villanova, PA 19085

USA

**Abstract:**

A happy number *N* is defined by the condition
*S*_{n}(*N*)= 1 for some
number *n* of iterations of the function *S*,
where *S*(*N*) is the sum
of the squares of the digits of *N*. Up to 10^{20}, the longest
known string of consecutive happy numbers was length five. We find the
smallest string of consecutive happy numbers of length 6, 7, 8, ...,
13. For instance, the smallest string of six consecutive happy numbers
begins with *N* = 7899999999999959999999996. We also find the smallest
sequence of 3-consecutive cubic happy numbers of lengths 4, 5, 6, 7, 8,
and 9.

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(Concerned with sequence
A055629.)

Received August 26 2009;
revised version received November 27 2009;
May 4 2010; June 4 2010.
Published in *Journal of Integer Sequences*, June 8 2010.

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