A107329 Triangle read by rows: T(n,k) gives number of partitions of k, (k=1..n) into the prime factors of n, for n>=1.

0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Author

Wouter Meeussen, May 22 2005

Keywords

• easy
• nonn
• look
• tabl

Comments

T(n,n) equals A066882(n).

Examples

			T(30,12)=5 counting [2,2,2,2,2,2], [2,2,2,3,3], [3,3,3,3], [2,2,3,5] and [2,5,5].
Triangle begins:
  {0},
  {0, 1},
  {0, 0, 1},
  {0, 1, 0, 1},
  {0, 0, 0, 0, 1},
  {0, 1, 1, 1, 1, 2},
  {0, 0, 0, 0, 0, 0, 1},
  {0, 1, 0, 1, 0, 1, 0, 1},
  {0, 0, 1, 0, 0, 1, 0, 0, 1},
  ...
		

Links

• Alois P. Heinz, rows n = 1..200, flattened

Crossrefs

Cf. A066882.

Row sums +1 give A092976.

Programs

Maple

with(numtheory):
T:= proc(n) local b, l; l:= sort([factorset(n)[]]):
      b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
            b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
          end; forget(b):
      seq(b(k, nops(l)), k=1..n)
    end:
seq(T(n), n=1..20);  # Alois P. Heinz, Oct 28 2021

Mathematica

Table[Rest@CoefficientList[Series[1/Times @@ ((1-x^#)& /@ (First /@ FactorInteger[n])), {x, 0, n}], x], {n, 2, 24}]

Formula

T(n,k) is coefficient of x^k in 1/Product(1-x^p_i) with p_i the prime factors of n.

Extensions

T(1,1) = 0 prepended by Michel Marcus, Oct 28 2021