Generalized Happy Numbers
Helen G. Grundman, E. A. Teeple · The Fibonacci Quarterly · 2001
Let S2: Z +-> Z + denote the function that takes a positive integer to the sum of the squares of its digits. More generally, for e> 2 and 0 0 such that S2(a) = 1. For example, 13 is a happy number since *Sf (13) = 1. Notice that 4 is not a happy number. Its S2-sequence is periodic with 5f (4) = 4. It is simple to verify that every positive integer less than 100 either is a happy number or has an $2-sequence that enters the cyclic £2-sequence of 4. It can further be shown that, for each positive integer a> 100, S2(a) 0 such that S2(a) = 1 or 4. Generalizing the concept of a happy number, we say that a positive integer a is a cubic happy number if its ^-sequence eventually reaches 1. We note that a positive integer can be a cubic happy number only if it is congruent to 1 modulo 3. This follows immediately from the following lemma. Lemma 2: Given a e Z +, for all m, $3 m (a) = a (mod 3). Proof: Let a- Zf=0a|.10 / , 0 < at < 9. Using the fact that, for each i, af = at (mod 3) and Iff = 1 (mod 3), we get V=o J /=o /=o /=o Thus, by a simple induction argument, we get that, for all M G Z +, S™(a) = a (mod 3). • The fixed points and cycles of S3 are characterized in Theorem 3, which can be found without proof in [1].