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A325238
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First positive integer with each omega-sequence.
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34
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1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 96, 120, 128, 192, 210, 216, 240, 256, 360, 384, 420, 480, 512, 720, 768, 840, 900, 960, 1024, 1260, 1296, 1440, 1536, 1680, 1920, 2048, 2310, 2520, 2880, 3072, 3360, 3840, 4096, 4620, 5040, 5760, 6144, 6720
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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The sequence of terms together with their omega-sequences begins:
1:
2: 1
4: 2 1
6: 2 2 1
8: 3 1
12: 3 2 2 1
16: 4 1
24: 4 2 2 1
30: 3 3 1
32: 5 1
36: 4 2 1
48: 5 2 2 1
60: 4 3 2 2 1
64: 6 1
96: 6 2 2 1
120: 5 3 2 2 1
128: 7 1
192: 7 2 2 1
210: 4 4 1
216: 6 2 1
240: 6 3 2 2 1
256: 8 1
360: 6 3 3 1
384: 8 2 2 1
420: 5 4 2 2 1
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MATHEMATICA
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tomseq[n_]:=If[n<=1, {}, Most[FixedPointList[Sort[Length/@Split[#]]&, Sort[Last/@FactorInteger[n]]]]];
omseqs=Table[Total/@tomseq[n], {n, 1000}];
Sort[Table[Position[omseqs, x][[1, 1]], {x, Union[omseqs]}]]
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CROSSREFS
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Cf. A001221, A001222, A007916, A011784, A070175, A071625, A118914, A181819, A181821, A303555, A304465, A323014, A323023, A325238, A325239.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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