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%I #6 Apr 16 2019 15:26:05
%S 1,2,4,6,8,12,16,24,30,32,36,48,60,64,96,120,128,192,210,216,240,256,
%T 360,384,420,480,512,720,768,840,900,960,1024,1260,1296,1440,1536,
%U 1680,1920,2048,2310,2520,2880,3072,3360,3840,4096,4620,5040,5760,6144,6720
%N First positive integer with each omega-sequence.
%C We define the omega-sequence of n (row n of A323023) to have length A323014(n) = frequency depth of n, and the k-th part is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, given by red(n = p^i*...*q^j) = prime(i)*...*prime(j), i.e., the product of primes indexed by the prime exponents of n.
%e The sequence of terms together with their omega-sequences begins:
%e 1:
%e 2: 1
%e 4: 2 1
%e 6: 2 2 1
%e 8: 3 1
%e 12: 3 2 2 1
%e 16: 4 1
%e 24: 4 2 2 1
%e 30: 3 3 1
%e 32: 5 1
%e 36: 4 2 1
%e 48: 5 2 2 1
%e 60: 4 3 2 2 1
%e 64: 6 1
%e 96: 6 2 2 1
%e 120: 5 3 2 2 1
%e 128: 7 1
%e 192: 7 2 2 1
%e 210: 4 4 1
%e 216: 6 2 1
%e 240: 6 3 2 2 1
%e 256: 8 1
%e 360: 6 3 3 1
%e 384: 8 2 2 1
%e 420: 5 4 2 2 1
%t tomseq[n_]:=If[n<=1,{},Most[FixedPointList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]]]]];
%t omseqs=Table[Total/@tomseq[n],{n,1000}];
%t Sort[Table[Position[omseqs,x][[1,1]],{x,Union[omseqs]}]]
%Y Cf. A001221, A001222, A007916, A011784, A070175, A071625, A118914, A181819, A181821, A303555, A304465, A323014, A323023, A325238, A325239.
%K nonn
%O 1,2
%A _Gus Wiseman_, Apr 14 2019
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