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%I #39 May 29 2024 07:05:45
%S 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 27,34,29,30,31,33,8,35,16,39,32,45,37,36,41,38,43,40,47,42,53,44,49,
%U 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 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 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 The sequence is conjectured to be a permutation of the positive integers.
%H Scott R. Shannon, <a href="/A372974/b372974.txt">Table of n, a(n) for n = 1..10000</a>
%H Scott R. Shannon, <a href="/A372974/a372974_1.png">Image of the first 5000 terms</a>. The green line is a(n) = n.
%H 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 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 nn = 120; c[_] := False; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; j = 2; u = 3;
%t Do[k = u;
%t While[Or[GCD[j, k] > 1, PrimeNu[k] == #, c[k]] &[PrimeNu[j]], k++];
%t Set[{a[n], c[k], j}, {k, True, k}];
%t If[k == u, While[c[u], u++]], {n, 3, nn}];
%t Array[a, nn] (* _Michael De Vlieger_, May 28 2024 *)
%Y Cf. A001221, A027748, A372975, A352588, A109465.
%K nonn
%O 1,2
%A _Scott R. Shannon_, May 26 2024
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