Journal of Integer Sequences, Vol. 13 (2010), Article 10.6.3

Smallest Examples of Strings of Consecutive Happy Numbers

Robert Styer
Department of Mathematical Sciences
Villanova University
Villanova, PA 19085


A happy number N is defined by the condition Sn(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 1020, 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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