editing

approved

Cf. A000430, A055932, A056239, A112798, A181819, A323023, A325247, A325257, A325260, A325261, A325277.

approved

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proposed

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proposed

Cf. A000430, A055932, A056239, A112798, A181819, A323023, A325247, A325257, A325260, A325261, A325277.

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

Omega-sequence statistics: A001222 (first omega), A001221 (second), A001222 (first omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length), A323022 (fourth/frequency depth), A323023 A325248 (allHeinz number).

allocated for Gus WisemanNumbers 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

1,2

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.

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

normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];

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

Select[Range[100], normQ[omseq[#]]&]

Positions of normal numbers (A055932) in A325248.

Cf. A000430, A055932, A056239, A112798, A181819, A325247, A325257, A325260, A325261, A325277.

Omega-sequence statistics: A001221 (second), A001222 (first), A071625 (third), A304465 (second-to-last), A323014 (length), A323022 (fourth), A323023 (all).

allocated

nonn

Gus Wiseman, Apr 16 2019

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editing

allocated for Gus Wiseman

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