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%I #18 Aug 21 2021 20:46:23
%S 1,1,2,3,1,4,3,5,1,7,2,5,4,7,3,8,1,9,2,11,1,13,2,15,4,9,5,7,6,11,3,13,
%T 4,11,5,8,7,9,8,11,7,10,9,11,10,13,5,16,1,17,2,19,1,23,2,25,1,27,2,29,
%U 1,31,2,32,3,16,7,12,11,13,6,17,3,19,4,17,5,19
%N Lexicographically earliest sequence of positive terms such that for any distinct i and j, lcm(a(i), a(i+1)) and lcm(a(j), a(j+1)) are distinct.
%C See A317025 for the corresponding LCM.
%C This sequence has similarities with A088177.
%C For any n > 0, let g(n) = gcd(a(n), a(n+1)); between 1 and 800000, the function g takes only 5 times a value other than 1.
%C For any n > 0 and prime number p, if p divides a(n+1), then the p-adic valuation of a(n+1) is strictly greater than the p-adic valuation of a(n).
%C This sequence contains infinitely many distinct values.
%C The first occurrence of a prime number p, if not preceded by 1, is followed by 1.
%C The first occurrence of a prime power k, if not preceded by a divisor of k, is followed by 1.
%C If this sequence contains infinitely many 1's, then A317025 is a permutation of the natural numbers.
%H Rémy Sigrist, <a href="/A317024/b317024.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A317024/a317024.gp.txt">PARI program for A317024</a>
%H Rémy Sigrist, <a href="/A317024/a317024.png">Scatterplot of the ordinal transform of the first 500000 terms</a>
%e The first terms, alongside lcm(a(n), a(n+1)), are:
%e n a(n) lcm(a(n), a(n+1))
%e -- ---- -----------------
%e 1 1 1
%e 2 1 2
%e 3 2 6
%e 4 3 3
%e 5 1 4
%e 6 4 12
%e 7 3 15
%e 8 5 5
%e 9 1 7
%e 10 7 14
%e 11 2 10
%e 12 5 20
%e 13 4 28
%e 14 7 21
%e 15 3 24
%e 16 8 8
%e 17 1 9
%e 18 9 18
%e 19 2 22
%e 20 11 11
%o (PARI) See Links section.
%Y Cf. A088177, A317025.
%K nonn
%O 1,3
%A _Rémy Sigrist_, Jul 19 2018