Experimental Mathematics

The EKG Sequence

J. C. Lagarias, E. M. Rains, and N. J. A. Sloane

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Abstract

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

Article information

Source
Experiment. Math., Volume 11, Issue 3 (2002), 437-446.

Dates
First available in Project Euclid: 9 July 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1057777433

Mathematical Reviews number (MathSciNet)
MR1959753

Zentralblatt MATH identifier
1117.11302

Subjects
Primary: 11Bxx: Sequences and sets 11B83: Special sequences and polynomials 11B75: Other combinatorial number theory
Secondary: 11N36: Applications of sieve methods

Keywords
Electrocardiagram sequence EKG sequence

Citation

Lagarias, J. C.; Rains, E. M.; Sloane, N. J. A. The EKG Sequence. Experiment. Math. 11 (2002), no. 3, 437--446. https://projecteuclid.org/euclid.em/1057777433


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