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.
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 16 2019
STATUS
approved