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On consecutive happy numbers

Hao Pan
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Abstract

Let e>=1 and b>=2 be integers. For a positive integer n=\sum_{j=0}^ka_jb^j with 0<=a_j =0, where T_{e,b}^r is the r-th iteration of T_{e,b}. In this paper, we prove that there exist arbitrarily long sequences of consecutive (e,b)-happy numbers provided that e-1 is not divisible by p-1 for any prime divisor p of b-1.

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Hao Pan. 2006-07-27. On consecutive happy numbers. https://arxiv.org/abs/math/0607213

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