OFFSET
0,2
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).
EXAMPLE
Triangle begins:
{}
{}
1
2 2 1
4 2 2 1
5 3 2 2 1
7 3 3 1
8 4 3 2 2 1
11 4 3 2 2 1
13 4 3 2 2 1
15 4 4 1
16 5 4 2 2 1
19 5 4 2 2 1
20 6 4 2 2 1
22 6 4 2 1
24 6 5 2 2 1
28 6 5 2 2 1
29 7 5 2 2 1
32 7 5 2 2 1
33 8 5 2 2 1
36 8 5 2 2 1
38 8 5 2 2 1
40 8 6 2 2 1
41 9 6 2 2 1
45 9 6 2 2 1
47 9 6 2 2 1
49 9 6 3 2 2 1
52 9 6 3 2 2 1
55 9 6 3 2 2 1
56 10 6 3 2 2 1
59 10 6 3 2 2 1
MATHEMATICA
omseq[n_Integer]:=If[n<=1, {}, Total/@NestWhileList[Sort[Length/@Split[#]]&, Sort[Last/@FactorInteger[n]], Total[#]>1&]];
Table[omseq[n!], {n, 0, 30}]
CROSSREFS
Row lengths are A325272. Row sums are A325274. Row n is row A325275(n) of A112798. Second-to-last column is A325273. Column k = 1 is A022559. Column k = 2 is A000720. Column k = 3 is A071626.
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Apr 18 2019
STATUS
approved