Smallest examples of strings of consecutive happy numbers
J. Integer Seq. 13(6), Article ID 10.6.3, 10 p., electronic only (2010)
Summary
Summary: 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,$ \dots $, 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.
