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A325261
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Numbers whose omega-sequence does not cover an initial interval of positive integers.
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2
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8, 16, 24, 27, 30, 32, 36, 40, 42, 48, 54, 56, 64, 66, 70, 72, 78, 80, 81, 88, 96, 100, 102, 104, 105, 108, 110, 112, 114, 120, 125, 128, 130, 135, 136, 138, 144, 152, 154, 160, 162, 165, 168, 170, 174, 176, 180, 182, 184, 186, 189, 190, 192, 195, 196, 200
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OFFSET
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1,1
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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) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1).
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LINKS
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EXAMPLE
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The sequence of terms together with their omega sequences begins:
8: 3->1 108: 5->2->2->1 189: 4->2->2->1
16: 4->1 110: 3->3->1 190: 3->3->1
24: 4->2->2->1 112: 5->2->2->1 192: 7->2->2->1
27: 3->1 114: 3->3->1 195: 3->3->1
30: 3->3->1 120: 5->3->2->2->1 196: 4->2->1
32: 5->1 125: 3->1 200: 5->2->2->1
36: 4->2->1 128: 7->1 208: 5->2->2->1
40: 4->2->2->1 130: 3->3->1 210: 4->4->1
42: 3->3->1 135: 4->2->2->1 216: 6->2->1
48: 5->2->2->1 136: 4->2->2->1 222: 3->3->1
54: 4->2->2->1 138: 3->3->1 224: 6->2->2->1
56: 4->2->2->1 144: 6->2->2->1 225: 4->2->1
64: 6->1 152: 4->2->2->1 230: 3->3->1
66: 3->3->1 154: 3->3->1 231: 3->3->1
70: 3->3->1 160: 6->2->2->1 232: 4->2->2->1
72: 5->2->2->1 162: 5->2->2->1 238: 3->3->1
78: 3->3->1 165: 3->3->1 240: 6->3->2->2->1
80: 5->2->2->1 168: 5->3->2->2->1 243: 5->1
81: 4->1 170: 3->3->1 246: 3->3->1
88: 4->2->2->1 174: 3->3->1 248: 4->2->2->1
96: 6->2->2->1 176: 5->2->2->1 250: 4->2->2->1
100: 4->2->1 180: 5->3->2->2->1 252: 5->3->2->2->1
102: 3->3->1 182: 3->3->1 255: 3->3->1
104: 4->2->2->1 184: 4->2->2->1 256: 8->1
105: 3->3->1 186: 3->3->1 258: 3->3->1
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MATHEMATICA
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normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];
omseq[n_Integer]:=If[n<=1, {}, Total/@NestWhileList[Sort[Length/@Split[#]]&, Sort[Last/@FactorInteger[n]], Total[#]>1&]];
Select[Range[100], !normQ[omseq[#]]&]
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CROSSREFS
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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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