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A325261 Numbers whose omega-sequence does not cover an initial interval of positive integers. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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).

LINKS

Table of n, a(n) for n=1..56.

EXAMPLE

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

MATHEMATICA

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[#]]&]

CROSSREFS

Complement of A325251.

Cf. A000430, A055932, A056239, A112798, A181819, A323023, A325247, A325257, A325260, A325262, A325277.

Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).

Sequence in context: A188438 A282256 A037370 * A229972 A144591 A078131

Adjacent sequences:  A325258 A325259 A325260 * A325262 A325263 A325264

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 23 2019

STATUS

approved

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Last modified August 21 19:04 EDT 2019. Contains 326168 sequences. (Running on oeis4.)