A325411 Numbers whose omega-sequence has repeated parts.

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 102, 104, 105

Offset: 1

Author

Gus Wiseman, Apr 24 2019

Keywords

• nonn

Comments

First differs from A323304 in lacking 216. First differs from A106543 in having 144.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose omega-sequence has repeated parts. The enumeration of these partitions by sum is given by A325285.

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), which has repeated parts, so 180 is in the sequence.

Examples

			The sequence of terms together with their omega-sequences begins:
   6: 2 2 1       51: 2 2 1         86: 2 2 1        119: 2 2 1
  10: 2 2 1       52: 3 2 2 1       87: 2 2 1        120: 5 3 2 2 1
  12: 3 2 2 1     54: 4 2 2 1       88: 4 2 2 1      122: 2 2 1
  14: 2 2 1       55: 2 2 1         90: 4 3 2 2 1    123: 2 2 1
  15: 2 2 1       56: 4 2 2 1       91: 2 2 1        124: 3 2 2 1
  18: 3 2 2 1     57: 2 2 1         92: 3 2 2 1      126: 4 3 2 2 1
  20: 3 2 2 1     58: 2 2 1         93: 2 2 1        129: 2 2 1
  21: 2 2 1       60: 4 3 2 2 1     94: 2 2 1        130: 3 3 1
  22: 2 2 1       62: 2 2 1         95: 2 2 1        132: 4 3 2 2 1
  24: 4 2 2 1     63: 3 2 2 1       96: 6 2 2 1      133: 2 2 1
  26: 2 2 1       65: 2 2 1         98: 3 2 2 1      134: 2 2 1
  28: 3 2 2 1     66: 3 3 1         99: 3 2 2 1      135: 4 2 2 1
  30: 3 3 1       68: 3 2 2 1      102: 3 3 1        136: 4 2 2 1
  33: 2 2 1       69: 2 2 1        104: 4 2 2 1      138: 3 3 1
  34: 2 2 1       70: 3 3 1        105: 3 3 1        140: 4 3 2 2 1
  35: 2 2 1       72: 5 2 2 1      106: 2 2 1        141: 2 2 1
  38: 2 2 1       74: 2 2 1        108: 5 2 2 1      142: 2 2 1
  39: 2 2 1       75: 3 2 2 1      110: 3 3 1        143: 2 2 1
  40: 4 2 2 1     76: 3 2 2 1      111: 2 2 1        144: 6 2 2 1
  42: 3 3 1       77: 2 2 1        112: 5 2 2 1      145: 2 2 1
  44: 3 2 2 1     78: 3 3 1        114: 3 3 1        146: 2 2 1
  45: 3 2 2 1     80: 5 2 2 1      115: 2 2 1        147: 3 2 2 1
  46: 2 2 1       82: 2 2 1        116: 3 2 2 1      148: 3 2 2 1
  48: 5 2 2 1     84: 4 3 2 2 1    117: 3 2 2 1      150: 4 3 2 2 1
  50: 3 2 2 1     85: 2 2 1        118: 2 2 1        152: 4 2 2 1
		

Crossrefs

Positions of nonsquarefree numbers in A325248.

Cf. A056239, A112798, A118914, A181819, A323023, A325247, A325249, A325250, A325251, A325277, A325285.

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

Programs

Mathematica

omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
Select[Range[100],!UnsameQ@@omseq[#]&]