cp's OEIS Frontend

A317024 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.

%I A317024 #18 Aug 21 2021 20:46:23
%S A317024 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 A317024 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 A317024 1,31,2,32,3,16,7,12,11,13,6,17,3,19,4,17,5,19
%N A317024 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 A317024 See A317025 for the corresponding LCM.
%C A317024 This sequence has similarities with A088177.
%C A317024 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 A317024 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 A317024 This sequence contains infinitely many distinct values.
%C A317024 The first occurrence of a prime number p, if not preceded by 1, is followed by 1.
%C A317024 The first occurrence of a prime power k, if not preceded by a divisor of k, is followed by 1.
%C A317024 If this sequence contains infinitely many 1's, then A317025 is a permutation of the natural numbers.
%H A317024 Rémy Sigrist, <a href="/A317024/b317024.txt">Table of n, a(n) for n = 1..10000</a>
%H A317024 Rémy Sigrist, <a href="/A317024/a317024.gp.txt">PARI program for A317024</a>
%H A317024 Rémy Sigrist, <a href="/A317024/a317024.png">Scatterplot of the ordinal transform of the first 500000 terms</a>
%e A317024 The first terms, alongside lcm(a(n), a(n+1)), are:
%e A317024   n  a(n)  lcm(a(n), a(n+1))
%e A317024   -- ----  -----------------
%e A317024    1    1    1
%e A317024    2    1    2
%e A317024    3    2    6
%e A317024    4    3    3
%e A317024    5    1    4
%e A317024    6    4   12
%e A317024    7    3   15
%e A317024    8    5    5
%e A317024    9    1    7
%e A317024   10    7   14
%e A317024   11    2   10
%e A317024   12    5   20
%e A317024   13    4   28
%e A317024   14    7   21
%e A317024   15    3   24
%e A317024   16    8    8
%e A317024   17    1    9
%e A317024   18    9   18
%e A317024   19    2   22
%e A317024   20   11   11
%o A317024 (PARI) See Links section.
%Y A317024 Cf. A088177, A317025.
%K A317024 nonn
%O A317024 1,3
%A A317024 _Rémy Sigrist_, Jul 19 2018