A318950 Regular triangle where T(n,k) is the number of factorizations of n into factors > 1 with sum k.

0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Oct 22 2018

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  0
  0 1
  0 0 1
  0 0 0 2
  0 0 0 0 1
  0 0 0 0 1 1
  0 0 0 0 0 0 1
  0 0 0 0 0 2 0 1
  0 0 0 0 0 1 0 0 1
  0 0 0 0 0 0 1 0 0 1
  0 0 0 0 0 0 0 0 0 0 1
  0 0 0 0 0 0 2 1 0 0 0 1
  0 0 0 0 0 0 0 0 0 0 0 0 1
  0 0 0 0 0 0 0 0 1 0 0 0 0 1
  0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
  0 0 0 0 0 0 0 3 0 1 0 0 0 0 0 1
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
  0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
  0 0 0 0 0 0 0 0 2 0 0 1 0 0 0 0 0 0 0 1
Row 12 {0,0,0,0,0,0,2,1,0,0,0,1} corresponds to the factorizations:
  . . . . . . (3*4)   (2*6) . . . (12)
              (2*2*3)
		

Crossrefs

Row sums are A001055. Column sums are A002865.

Cf. A069016, A096276, A301987, A319000, A319005, A319057, A319916.

Programs

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],Total[#]==k&]],{n,20},{k,n}]