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The EKG Sequence

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.

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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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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
Record Number:CaltechAUTHORS:20171011-152214607
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:82295
Deposited By: Tony Diaz
Deposited On:12 Oct 2017 16:22
Last Modified:15 Nov 2021 19:49

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