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A363956 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that is a multiple of prime(omega(a(n-1))).

%I A363956 #10 Jul 01 2023 09:14:52
%S A363956 1,2,4,6,3,8,10,9,12,15,18,21,24,27,14,30,5,16,20,33,36,39,42,25,22,
%T A363956 45,48,51,54,57,60,35,63,66,40,69,72,75,78,50,81,26,84,55,87,90,65,93,
%U A363956 96,99,102,70,80,105,85,108,111,114,95,117,120,100,123,126,110,115,129,132,125,28,135,138
%N A363956 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that is a multiple of prime(omega(a(n-1))).
%C A363956 The sequence is conjectured to be a permutation of the positive integers, although it takes many terms for the primes to appear, e.g., a(210667) = 17. After the first two terms the only fixed points in the first 500000 terms are 15777 and 86977, although it is possible more exist.
%H A363956 Michael De Vlieger, <a href="/A363956/b363956.txt">Table of n, a(n) for n = 1..10000</a>
%H A363956 Scott R. Shannon, <a href="/A363956/a363956.png">Image of the first 500000 terms</a>. The green line is a(n) = n.
%H A363956 Michael De Vlieger, <a href="/A363956/a363956_1.png">Log log plot of a(n)</a>, n = 1..2^20.
%H A363956 Michael De Vlieger, <a href="/A363956/a363956_2.png">Log log plot of a(n)</a>, n = 1..2^12, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor composite in blue. Powerful numbers that are not prime powers are highlighted in light blue.
%e A363956 a(10) = 15 as prime(omega(a(9))) = prime(A001221(12)) = prime(2) = 3, and 15 is the smallest unused number that is a multiple of 3.
%t A363956 nn = 120; c[_] := False; m[_] := 1; f[x_] := Prime@ PrimeNu[x];
%t A363956 Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 2]; j = 2;
%t A363956 Do[k = f[j];
%t A363956  While[c[m[k] k], m[k]++]; k *= m[k];
%t A363956  Set[{a[n], c[k], j}, {k, True, k}], {n, 3, nn}];
%t A363956 Array[a, nn] (* _Michael De Vlieger_, Jul 01 2023 *)
%Y A363956 Cf. A363504, A001221, A000040, A001222, A351495.
%K A363956 nonn
%O A363956 1,2
%A A363956 _Scott R. Shannon_, Jun 29 2023