A379737 Triangle read by rows where, for n >= k, T(n,k) is the number of factorizations of n into distinct factors > 1 with sum k.

0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 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, 1, 1, 0, 0, 0, 1

Offset

1

Links

Table of n, a(n) for n=1..78.

Formula

For k >= n we have T(k,n) = A379678(n,k).

Example

The T(72,17) = 2 factorizations are: (2*3*12), (8*9).
Triangle begins:
  0
  0 1
  0 0 1
  0 0 0 1
  0 0 0 0 1
  0 0 0 0 1 1
  0 0 0 0 0 0 1
  0 0 0 0 0 1 0 1
  0 0 0 0 0 0 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 1 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 0 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 0 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 1 0 0 1 0 0 0 0 0 0 0 1
For example, row n = 12 counts the following strict factorizations:
   k=1 k=2 k=3 k=4 k=5 k=6  k=7   k=8  k=9 k=10 k=11 k=12
    .   .   .   .   .   .  (3*4) (2*6)  .   .    .   (12)

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], UnsameQ@@#&&Total[#]==k&]], {n, 30}, {k, n}]

Crossrefs

Column sums are A025147 = strict partitions into parts > 1, non-strict A002865.

Row sums are A045778 = strict factorizations, non-strict A001055.

The non-strict version is A318950.

Extend to an array (padded with zeros) and take transpose to get A379678.

A000041 counts integer partitions, strict A000009.

A316439 counts factorizations by length, partitions A008284.

A326622 counts factorizations with integer mean, strict A328966.

Arrays counting multisets by sum and product:

- partitions: A379666, antidiagonal sums A379667

- without ones: A379668, antidiagonal sums A379669 (zeros A379670)

- strict partitions: A379671, antidiagonal sums A379672

- strict without ones: A379678, antidiagonal sums A379679 (zeros A379680)

Counting and ranking multisets by comparing sum and product:

- same: A001055, ranks A301987

- divisible: A057567, ranks A326155

- divisor: A057568, ranks A326149, see A326156, A326172, A379733

- greater than: A096276 shifted right, ranks A325038

- greater or equal: A096276, ranks A325044

- less than: A114324, ranks A325037, see A318029

- less or equal: A319005, ranks A379721

- different: A379736, ranks A379722, see A111133

Cf. A028422, A069016, A319000, A319057, A319916, A325036, A326152, A379720.

Sequence in context: A080887 A099395 A182581 * A288203 A238470 A286748

Adjacent sequences: A379728 A379729 A379732 * A379738 A379741 A379742

Keyword

nonn,tabl,new

Author

Gus Wiseman, Jan 03 2025

Status

approved