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

1,10

Links

Table of n, a(n) for n=1..87.

Example

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)

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}]

Crossrefs

Row sums are A001055. Column sums are A002865.

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

Sequence in context: A138088 A112765 A105966 * A319000 A083915 A083892

Adjacent sequences: A318947 A318948 A318949 * A318951 A318952 A318953

Keyword

nonn,tabl

Author

Gus Wiseman, Oct 22 2018

Status

approved