Geodesic
Smallest examples of strings of consecutive happy numbers
Journal of integer sequences, Tome 13 (2010) no. 6
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.
Classification : 11A63
Keywords: happy number, cubic happy number 
@article{JIS_2010__13_6_a4,
     author = {Styer,  Robert},
     title = {Smallest examples of strings of consecutive happy numbers},
     journal = {Journal of integer sequences},
     year = {2010},
     volume = {13},
     number = {6},
     zbl = {1238.11007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2010__13_6_a4/}
}
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Styer,  Robert. Smallest examples of strings of consecutive happy numbers. Journal of integer sequences, Tome 13 (2010) no. 6. http://geodesic.mathdoc.fr/item/JIS_2010__13_6_a4/