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 → ∞; 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 c1n ≤ a(n) ≤ c2n for constants c1, c2. There remains a large gap between what is conjectured and what is proved.