A325261 Numbers whose omega-sequence does not cover an initial interval of positive integers.

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

Offset: 1

Author

Gus Wiseman, Apr 23 2019

Keywords

• nonn

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).

Examples

			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
		

Crossrefs

Complement of A325251.

Cf. A000430, A055932, A056239, A112798, A181819, A323023, A325247, 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).

Programs

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[#]]&]