Geodesic
Sequences of generalized happy numbers with small bases
Journal of integer sequences, Tome 10 (2007) no. 1
For bases $b \le 5$ and exponents $e \ge 2$, there exist arbitrarily long finite sequences of $d$-consecutive $e$-power $b$-happy numbers for a specific $d = d(e,b)$, which is shown to be minimal possible.
Keywords: happy numbers, consecutive, base 
@article{JIS_2007__10_1_a5,
     author = {Grundman,  H.G. and Teeple,  E.A.},
     title = {Sequences of generalized happy numbers with small bases},
     journal = {Journal of integer sequences},
     year = {2007},
     volume = {10},
     number = {1},
     zbl = {1115.11009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2007__10_1_a5/}
}
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Grundman,  H.G.; Teeple,  E.A. Sequences of generalized happy numbers with small bases. Journal of integer sequences, Tome 10 (2007) no. 1. http://geodesic.mathdoc.fr/item/JIS_2007__10_1_a5/