Lagarias, J. C.; Rains, E. M.; Sloane, N. J. A. The EKG sequence. (English) Zbl 1117.11302 Exp. Math. 11, No. 3, 437-446 (2002). Summary: The EKG or electrocardiogram sequence is defined by \(a(1) = 1\), \(a(2) =2\) and, for \(n \geq 3\), \(a(n)\) 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 \[ a(n) = n (1+ 1/(3 \log n)) + o(n/ \log n)\;\text{as}\;n \to \infty; \] 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 \(\{a(n): n \geq 1 \}\) is a permutation of the natural numbers and that \(c_1 n \leq a (n) \leq c_2 n\) for constants \(c_1, c_2\). There remains a large gap between what is conjectured and what is proved. Cited in 1 ReviewCited in 2 Documents MSC: 11B83 Special sequences and polynomials 11B75 Other combinatorial number theory 11N36 Applications of sieve methods Keywords:Electrocardiagram sequence; EKG sequence Software:OEIS × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid EuDML Link Online Encyclopedia of Integer Sequences: At step n in computing A064413, sequence gives smallest multiple of 2 not yet seen, divided by 2. At step n in computing A064413, sequence gives smallest multiple of 3 not yet seen, divided by 3. At step n in computing A064413, sequence gives smallest multiple of 5 not yet seen, divided by 5. a(1) = 1, a(2) = 2, a(3) = 3; for n >3 a(n) = smallest number not already used such that gcd(a(n), a(n-1)) >= 3. a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 4; for n > 4, a(n) = smallest number not already used such that gcd(a(n), a(n-1)) >= 4. a(n) = n for n <= 5; for n > 5, a(n) = smallest number not already used such that gcd(a(n), a(n-1)) >= 5. Numbers n such that A064413(n+1) = n. Probably complete. Term at which n appears in A064413 (if it begins at 2). Record high values in A064413. First differences of A064413. Positions of powers of 2 in A064413 (if it starts at 2). The even subsequence of A064413. The even subsequence of A064413, divided by 2. The odd subsequence of A064413. Where the even terms appear in A064413 (if it begins with 2). Where the odd terms appear in A064413 (if it begins at 2). Number of even terms among first n terms of A064413. Number of odd terms among first n terms of A064413 (if it begins at 2). Length of n-th run of evens or odds in A064413. Length of n-th run of even numbers in A064413. Length of n-th run of odd numbers in A064413. Regard A064413 as giving a permutation of the positive integers; sequence gives inverse permutation. Regard A064413 as giving a permutation of the positive integers; sequence gives first infinite cycle, beginning at its smallest term, 7. Regard A064413 as giving a permutation of the positive integers; sequence gives first infinite cycle, reading backwards beginning at its smallest term, 7. Regard A064413 as giving a permutation of the positive integers; sequence gives second infinite cycle, beginning at its smallest term, 73. Regard A064413 as giving a permutation of the positive integers; sequence gives second infinite cycle, reading backwards beginning at its smallest term, 73. Regard A064413 as giving a permutation of the positive integers; sequence gives (presumed) smallest term in each cycle of this permutation. Smallest controlling prime when A064413(n) is computed. A064413(n) written in base of primes, read from right to left, written as a string. A064413(n) written in base of primes, read from right to left, written as n-th row of a table. Lengths of the finite cycles in the permutation defined by A064413. Number of odd terms among first n terms of A064413. Where the even terms appear in A064413. Positions of powers of 2 in A064413. Position of n-th prime in A064413. Inverse permutation to A064417. Where the odd terms appear in A064413. Inverse permutation to A064418. Inverse permutation to A064419. GCD of consecutive members of the EKG sequence A064413. Numbers k such that the k-th term of the EKG sequence (A064413(k)) has more than one controlling prime. A064413^2, that is each term in the EKG sequence is squared. Cumulative number of tests performed to find the n-th term of the EKG sequence A064413. Numbers k such that A073734(k) is neither squarefree nor a prime power. References: [1] Ayres, J. Sept. 30 2001. Sept. 30, [Ayres 01], personal communication [2] Erdös P., Acta Math. Acad. Sci. Hungar. 41 pp 169– (1983) · Zbl 0518.10063 · doi:10.1007/BF01994075 [3] Hooley C., Applications of Sieve Methods to the Theory of Numbers (1976) · Zbl 0327.10044 [4] Sloane N. J. A., The On-Line Encyclopedia of Integer Sequences · Zbl 1044.11108 [5] Zagier D., Math. Intelligencer 0 pp 7– (1977) · Zbl 0392.10001 · doi:10.1007/BF03039306 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.