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

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

Offset: 0

Author

Gus Wiseman, Apr 18 2019

Keywords

• nonn
• tabf

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).

Examples

			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
		

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.

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

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).

Programs

Mathematica

omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
Table[omseq[n!],{n,0,30}]