cp's OEIS Frontend

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

%I A325261 #6 Aug 22 2019 09:54:18
%S A325261 8,16,24,27,30,32,36,40,42,48,54,56,64,66,70,72,78,80,81,88,96,100,
%T A325261 102,104,105,108,110,112,114,120,125,128,130,135,136,138,144,152,154,
%U A325261 160,162,165,168,170,174,176,180,182,184,186,189,190,192,195,196,200
%N A325261 Numbers whose omega-sequence does not cover an initial interval of positive integers.
%C A325261 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).
%e A325261 The sequence of terms together with their omega sequences begins:
%e A325261     8: 3->1           108: 5->2->2->1        189: 4->2->2->1
%e A325261    16: 4->1           110: 3->3->1           190: 3->3->1
%e A325261    24: 4->2->2->1     112: 5->2->2->1        192: 7->2->2->1
%e A325261    27: 3->1           114: 3->3->1           195: 3->3->1
%e A325261    30: 3->3->1        120: 5->3->2->2->1     196: 4->2->1
%e A325261    32: 5->1           125: 3->1              200: 5->2->2->1
%e A325261    36: 4->2->1        128: 7->1              208: 5->2->2->1
%e A325261    40: 4->2->2->1     130: 3->3->1           210: 4->4->1
%e A325261    42: 3->3->1        135: 4->2->2->1        216: 6->2->1
%e A325261    48: 5->2->2->1     136: 4->2->2->1        222: 3->3->1
%e A325261    54: 4->2->2->1     138: 3->3->1           224: 6->2->2->1
%e A325261    56: 4->2->2->1     144: 6->2->2->1        225: 4->2->1
%e A325261    64: 6->1           152: 4->2->2->1        230: 3->3->1
%e A325261    66: 3->3->1        154: 3->3->1           231: 3->3->1
%e A325261    70: 3->3->1        160: 6->2->2->1        232: 4->2->2->1
%e A325261    72: 5->2->2->1     162: 5->2->2->1        238: 3->3->1
%e A325261    78: 3->3->1        165: 3->3->1           240: 6->3->2->2->1
%e A325261    80: 5->2->2->1     168: 5->3->2->2->1     243: 5->1
%e A325261    81: 4->1           170: 3->3->1           246: 3->3->1
%e A325261    88: 4->2->2->1     174: 3->3->1           248: 4->2->2->1
%e A325261    96: 6->2->2->1     176: 5->2->2->1        250: 4->2->2->1
%e A325261   100: 4->2->1        180: 5->3->2->2->1     252: 5->3->2->2->1
%e A325261   102: 3->3->1        182: 3->3->1           255: 3->3->1
%e A325261   104: 4->2->2->1     184: 4->2->2->1        256: 8->1
%e A325261   105: 3->3->1        186: 3->3->1           258: 3->3->1
%t A325261 normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t A325261 omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
%t A325261 Select[Range[100],!normQ[omseq[#]]&]
%Y A325261 Complement of A325251.
%Y A325261 Cf. A000430, A055932, A056239, A112798, A181819, A323023, A325247, A325260, A325262, A325277.
%Y A325261 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).
%K A325261 nonn
%O A325261 1,1
%A A325261 _Gus Wiseman_, Apr 23 2019