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A325411 Numbers whose omega-sequence has repeated parts. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..65.

EXAMPLE

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

MATHEMATICA

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

Select[Range[100], !UnsameQ@@omseq[#]&]

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).

Sequence in context: A168638 A024619 A323304 * A106543 A324455 A327476

Adjacent sequences:  A325408 A325409 A325410 * A325412 A325413 A325414

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 24 2019

STATUS

approved

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Last modified November 19 06:03 EST 2019. Contains 329310 sequences. (Running on oeis4.)