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

1,2

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.

Links

Table of n, a(n) for n=1..67.

Example

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

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

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).

Sequence in context: A342521 A078129 A325368 * A330977 A188437 A344414

Adjacent sequences: A325248 A325249 A325250 * A325252 A325253 A325254

Keyword

nonn

Author

Gus Wiseman, Apr 16 2019

Status

approved