cp's OEIS Frontend

A372974 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that a(n) is coprime to a(n-1) and omega(a(n)) does not equal omega(a(n-1)).

%I A372974 #39 May 29 2024 07:05:45
%S A372974 1,2,15,4,21,5,6,7,10,3,14,9,20,11,12,13,18,17,22,19,24,23,26,25,28,
%T A372974 27,34,29,30,31,33,8,35,16,39,32,45,37,36,41,38,43,40,47,42,53,44,49,
%U A372974 46,59,48,61,50,67,51,64,55,71,52,73,54,79,56,81,58,83,57,70,69,89,60,77,78,85,66,65,84
%N A372974 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that a(n) is coprime to a(n-1) and omega(a(n)) does not equal omega(a(n-1)).
%C A372974 The fixed points show an unusual pattern; they begin 1, 2, 4, 69, 190, 438, 545, 725, 732, 909 and it appears, based on a graph of the sequence (see the attached image of the first 5000 terms) there may be no more. However more exist at 324388, 330574, 333069, 333531,..., 369752. Then once again there is a large gap until 2704713, 2726054, 2760963, ... . It is unclear what causes this behavior.
%C A372974 The sequence is conjectured to be a permutation of the positive integers.
%H A372974 Scott R. Shannon, <a href="/A372974/b372974.txt">Table of n, a(n) for n = 1..10000</a>
%H A372974 Scott R. Shannon, <a href="/A372974/a372974_1.png">Image of the first 5000 terms</a>. The green line is a(n) = n.
%H A372974 Michael De Vlieger, <a href="/A372974/a372974_2.png">Log log scatterplot of a(n)</a>, n = 1..2^16, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue or purple, with purple indicating powerful numbers that are not prime powers.
%e A372974 a(3) = 15 as a(2) = 2 and omega(2) = A001221(2) = 1, and 15 is coprime to 2 while omega(15) = A001221(15) = 2 which does not equal 1. No smaller number satisfies both of these requirements.
%t A372974 nn = 120; c[_] := False; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; j = 2; u = 3;
%t A372974 Do[k = u;
%t A372974   While[Or[GCD[j, k] > 1, PrimeNu[k] == #, c[k]] &[PrimeNu[j]], k++];
%t A372974   Set[{a[n], c[k], j}, {k, True, k}];
%t A372974   If[k == u, While[c[u], u++]], {n, 3, nn}];
%t A372974 Array[a, nn] (* _Michael De Vlieger_, May 28 2024 *)
%Y A372974 Cf. A001221, A027748, A372975, A352588, A109465.
%K A372974 nonn
%O A372974 1,2
%A A372974 _Scott R. Shannon_, May 26 2024