The EKG Sequence

Abstract

The EKC 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/(3logn)) + 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 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.

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