A325251 Numbers whose omega-sequence covers an initial interval of positive integers.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 82, 83, 84, 85, 86

Offset: 1

Author

Gus Wiseman, Apr 16 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).

The enumeration of these partitions by sum is given by A325260.

Examples

			The sequence of terms together with their omega sequences begins:
   1:              31: 1             63: 3 2 2 1
   2: 1            33: 2 2 1         65: 2 2 1
   3: 1            34: 2 2 1         67: 1
   4: 2 1          35: 2 2 1         68: 3 2 2 1
   5: 1            37: 1             69: 2 2 1
   6: 2 2 1        38: 2 2 1         71: 1
   7: 1            39: 2 2 1         73: 1
   9: 2 1          41: 1             74: 2 2 1
  10: 2 2 1        43: 1             75: 3 2 2 1
  11: 1            44: 3 2 2 1       76: 3 2 2 1
  12: 3 2 2 1      45: 3 2 2 1       77: 2 2 1
  13: 1            46: 2 2 1         79: 1
  14: 2 2 1        47: 1             82: 2 2 1
  15: 2 2 1        49: 2 1           83: 1
  17: 1            50: 3 2 2 1       84: 4 3 2 2 1
  18: 3 2 2 1      51: 2 2 1         85: 2 2 1
  19: 1            52: 3 2 2 1       86: 2 2 1
  20: 3 2 2 1      53: 1             87: 2 2 1
  21: 2 2 1        55: 2 2 1         89: 1
  22: 2 2 1        57: 2 2 1         90: 4 3 2 2 1
  23: 1            58: 2 2 1         91: 2 2 1
  25: 2 1          59: 1             92: 3 2 2 1
  26: 2 2 1        60: 4 3 2 2 1     93: 2 2 1
  28: 3 2 2 1      61: 1             94: 2 2 1
  29: 1            62: 2 2 1         95: 2 2 1
		

Crossrefs

Positions of normal numbers (A055932) in A325248.

Cf. A000430, A055932, A056239, A112798, A181819, A323023, A325247, A325260, A325261, 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).

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[#]]&]