Lagarias, J. C. and Rains, E. M. and Sloane, N. J. A.
(2002)
*The EKG Sequence.*
Experimental Mathematics, 11
(3).
pp. 437-446.
ISSN 1058-6458.
doi:10.1080/10586458.2002.10504486.
https://resolver.caltech.edu/CaltechAUTHORS:20171011-152214607

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## 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.

Item Type: | Article | ||||||||||||
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Additional Information: | © 2002 A K Peters, Ltd. Received December 12, 2001; accepted in revised form March 11, 2002. We thank Jonathan Ayres for discovering this wonderful sequence. We also thank a referee for helpful comments. | ||||||||||||

Subject Keywords: | Electrocardiagrarn sequence, EKG sequence | ||||||||||||

Issue or Number: | 3 | ||||||||||||

Classification Code: | 2000 AMS Subject Classification: Primary 11Bxx, 11B83, 11B75. Secondary 11N36 | ||||||||||||

DOI: | 10.1080/10586458.2002.10504486 | ||||||||||||

Record Number: | CaltechAUTHORS:20171011-152214607 | ||||||||||||

Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20171011-152214607 | ||||||||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||

ID Code: | 82295 | ||||||||||||

Collection: | CaltechAUTHORS | ||||||||||||

Deposited By: | Tony Diaz | ||||||||||||

Deposited On: | 12 Oct 2017 16:22 | ||||||||||||

Last Modified: | 15 Nov 2021 19:49 |

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