Augmented generalized happy functions

An augmented generalized happy function, S[c,b] maps a positive integer to the sum of the squares of its base b digits and a non-negative integer c. A positive integer u is in a cycle of S[c,b] if, for some positive integer k, Sk [c,b](u) = u, and, for positive integers v and w, v is w-attracted for S[c,b] if, for some non-negative integer ℓ, Sℓ [c,b](v) = w. In this paper, we prove that, for each c 0 and b 2, and for any u in a cycle of S[c,b]: (1) if b is even, then there exist arbitrarily long sequences of consecutive u-attracted integers, and (2) if b is odd, then there exist arbitrarily long sequences of 2-consecutive u-attracted integers.