A316439 Irregular triangle where T(n,k) is the number of factorizations of n into k factors > 1, with k ranging from 1 to Omega(n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 3, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 3

Offset: 1

Author

Gus Wiseman, Jul 03 2018

Keywords

• nonn
• tabf

Examples

			The factorizations of 24 are (2*2*2*3), (2*2*6), (2*3*4), (2*12), (3*8), (4*6), (24) so the 24th row is {1, 3, 2, 1}.
Triangle begins:
  {}
  1
  1
  1  1
  1
  1  1
  1
  1  1  1
  1  1
  1  1
  1
  1  2  1
  1
  1  1
  1  1
  1  2  1  1
  1
  1  2  1
  1
  1  2  1
  1  1
  1  1
  1
  1  3  2  1
  1  1
  1  1
  1  1  1
  1  2  1
  1
  1  3  1
		

Links

• Alois P. Heinz, Rows n = 1..20000, flattened

Crossrefs

Cf. A001222 (row lengths), A001055 (row sums), A001970, A007716, A045778, A162247, A259936, A281116, A303386.

Programs

Maple

g:= proc(n, k) option remember; `if`(n>k, 0, x)+
      `if`(isprime(n), 0, expand(x*add(`if`(d>k, 0,
      g(n/d, d)), d=numtheory[divisors](n) minus {1, n})))
    end:
T:= n-> `if`(n=1, [][], (p-> seq(coeff(p, x, i)
        , i=1..degree(p)))(g(n$2))):
seq(T(n), n=1..50);  # Alois P. Heinz, Aug 11 2019

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Length[#]==k&]],{n,100},{k,PrimeOmega[n]}]