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

0,2

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).

Links

Table of n, a(n) for n=0..79.

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.

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).

Sequence in context: A174498 A146478 A289500 * A098804 A191320 A180228

Adjacent sequences: A325273 A325274 A325275 * A325277 A325278 A325279

Keyword

nonn,tabf

Author

Gus Wiseman, Apr 18 2019

Status

approved