Numbers and the heights of their happiness

Abstract

A generalized happy function, S e,b maps a positive integer to the sum of its base b digits raised to the e th power. We say that x is a base b, e power, height h, u attracted number if h is the smallest positive integer so that S h e,b (x) = u. Happy numbers are then base 10, 2 power, 1 attracted numbers of any height. Let σ h,e,b (u) denote the smallest height h, u attracted number for a fixed base b and exponent e and let g(e) denote the smallest number so that every integer can be written as x e 1 + x e 2 + ... + x e g(e) for some nonnegative integers x 1 , x 2 , ..., x g(e) . In this paper we prove that if p e,b is the smallest nonnegative integer such that b p e,b > g(e), d = g(e) + 1p e,b , and σ h,e,b (u) ≥ b d , then S e,b (σ h+1,e,b (u)) = σ h,e,b (u).1991 Mathematics Subject Classification. 11A63.

Happy Numbers, Happy Functions, and Their Variations: A Survey

“…For some work on parallel, more general questions (for example, with S e,b in place of S 2,10 ), see [43].…”

“…Can this answer be generalized to other bases and/or exponents? -Under what conditions are the digits of the least e-power b-happy number of a given height h a permutation of the digits of the second least e-power b-happy number of height h [43]? -Is there a generalized version of Theorem 14 with a bound given in terms of e and b [43]?…”

“…Mathematicians and computer scientists have been attracted by the difficult open problems, while more accessible problems have been used for undergraduate and Masters level research. (For examples of student research, see [28,41,43,55,56].) Though happy numbers have continued to appear in recreational and educational publications (and in a 2007 episode of Dr. Who), we will now focus on research publications.…”