The EKG Sequence
  The EKG or electrocardiogram sequence is defined by a(1) = 1, a(2) = 2 and, for n >= 3, a(n) is the smallest natural number not already in the sequence with the property that gcd {a(n-1), a(n)} > 1. In spite of its erratic local behavior, which when plotted resembles an electrocardiogram, its global behavior appears quite regular. We conjecture that almost all a(n) satisfy the asymptotic formula a(n) = n(1 + 1/(3 log n) + o(n/log n)) as n goes to infty; and that the exceptional values a(n) = p and a(n) = 3p, for p a prime, produce the spikes in the EKG sequence. We prove that {a(n): n >= 1} is a permutation of the natural numbers and that c_1 n <= a (n) <= c_2 n for constants c_1, c_2. There remains a large gap between what is conjectured and what is proved. 
J. C. Lagarias, E. M. Rains and N. J. A. Sloane