A happy number N is defined by the condition S_n(N)= 1 for some number n of iterations of the function S, where S(N) is the sum of the squares of the digits of N. Up to 10²⁰, the longest known string of consecutive happy numbers was length five. We find the smallest string of consecutive happy numbers of length 6, 7, 8, ..., 13. For instance, the smallest string of six consecutive happy numbers begins with N = 7899999999999959999999996. We also find the smallest sequence of 3-consecutive cubic happy numbers of lengths 4, 5, 6, 7, 8, and 9.