A369686 - OEIS

%I #30 Jul 18 2024 03:33:36

%S 1,2,3,5,2,1,1,7,3,2,1,1,11,1,1,1,1,5,1,1,1,1,1,2,1,1,1,1,1,7,13,3,1,

%T 1,1,1,1,2,1,1,1,1,1,17,1,1,19,1,1,1,1,1,1,1,23,1,1,1,1,1,1,1,1,1,1,

%U 29,1,1,1,1,1,1,1,11,1,5,1,1,1,1,1,1,3,31,1

%N LCM-transform of A359804 (see Comment and links).

%C Let b(k) be the Least Common Multiple (LCM) of the first k terms of A359804, then a(n) = b(n)/b(n-1), where sequence b(n) is A369685.

%C The property S (as defined in A368900) refers to what is observed in the positive integers (A000027), and also in the Doudna sequence (A005940), whereby each prime power appears prior to any of its multiples. The present sequence does not have this property since, for example, 26 = a(31) precedes 13 = a(42). Thus A369804 represents a significant disturbance of A000027 in that whereas it is conjectured to be a permutation of the positive integers, it does not preserve one of the basic properties of that sequence.

%H Michael De Vlieger, <a href="/A369686/b369686.txt">Table of n, a(n) for n = 1..10000</a>

%H Andrzej Nowicki, <a href="http://arxiv.org/abs/1310.2416">Strong divisibility and LCM-sequences</a>, arXiv:1310.2416 [math.NT], 2013.

%H Andrzej Nowicki, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.122.10.958">Strong divisibility and LCM-sequences</a>, Am. Math. Monthly (2015) Vol. 122, 958-966.

%F a(n) = A369685(n)/A369685(n-1).

%t nn = 120; c[_] = False; q[_] = 1;

%t Array[Set[{a[#], c[#]}, {#, True}] &, 2];

%t Set[{i, j}, {1, 2}]; m = 2; u = 3;

%t Do[

%t   (k = q[#]; While[c[k #], k++]; k *= #; While[c[# q[#]], q[#]++]) &[

%t   (p = 2; While[Divisible[i j, p], p = NextPrime[p]]; p)];

%t   Set[{a[n], c[k], i, j, m}, {#/m, True, j, k, #}] &[LCM[m, k]];

%t   If[k == u, While[c[u], u++]], {n, 3, nn}];

%t Array[a, nn] (* _Michael De Vlieger_, Jan 29 2024 *)

%Y Cf. A014963, A359804, A368900, A369685.

%K nonn

%O 1,2

%A _David James Sycamore_, Jan 28 2024

%E More terms from _Michael De Vlieger_, Jan 29 2024