On consecutive happy numbers

Abstract

AbstractLet e⩾1 and b⩾2 be integers. For a positive integer n=∑j=0kaj×bj with 0⩽aj<b, defineSe,b(n)=∑j=0kaje. n is called (e,b)-happy if Se,br(n)=1 for some r⩾0, where Se,br is the rth iteration of Se,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