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 A325276 Irregular triangle read by rows where row n is the omega-sequence of n!. 15

%I

%S 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,

%T 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,

%U 5,2,2,1,28,6,5,2,2,1,29,7,5,2,2,1

%N Irregular triangle read by rows where row n is the omega-sequence of n!.

%C 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).

%e Triangle begins:

%e {}

%e {}

%e 1

%e 2 2 1

%e 4 2 2 1

%e 5 3 2 2 1

%e 7 3 3 1

%e 8 4 3 2 2 1

%e 11 4 3 2 2 1

%e 13 4 3 2 2 1

%e 15 4 4 1

%e 16 5 4 2 2 1

%e 19 5 4 2 2 1

%e 20 6 4 2 2 1

%e 22 6 4 2 1

%e 24 6 5 2 2 1

%e 28 6 5 2 2 1

%e 29 7 5 2 2 1

%e 32 7 5 2 2 1

%e 33 8 5 2 2 1

%e 36 8 5 2 2 1

%e 38 8 5 2 2 1

%e 40 8 6 2 2 1

%e 41 9 6 2 2 1

%e 45 9 6 2 2 1

%e 47 9 6 2 2 1

%e 49 9 6 3 2 2 1

%e 52 9 6 3 2 2 1

%e 55 9 6 3 2 2 1

%e 56 10 6 3 2 2 1

%e 59 10 6 3 2 2 1

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

%t Table[omseq[n!],{n,0,30}]

%Y 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.

%Y Cf. A000142, A006939, A081401, A303555, A323023, A325238, A325275, A325277.

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

%K nonn,tabf

%O 0,2

%A _Gus Wiseman_, Apr 18 2019

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Last modified September 16 02:16 EDT 2019. Contains 327088 sequences. (Running on oeis4.)