2017/05/12 by Grundman, Helen G., Harris, Pamela E.
#11A63 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1705.04648
For b≤ -2 and e ≥ 2, let Se,b:ℤ→ℤ≥ 0 be the function taking an integer to the sum of the e-powers of the digits of its base b expansion. An integer a is a b-happy number if there exists k∈ℤ+ such that S2,bk(a) = 1. We prove that an integer is -2-happy if and only if it is congruent to 1 modulo 3 and that it is -3-happy if and only if it is odd. Defining a d-sequence to be an arithmetic sequence with constant difference d and setting d = gcd(2,b - 1), we prove that if b ≤ -3 odd or b ∈ \-4,-6,-8,-10\, there exist arbitrarily long finite sequences of d-consecutive b-happy numbers.