A212219 Number of representations of n as a sum of products of distinct pairs of positive integers >=2, n = Sum_{k=1..m} i_k*j_k with 2<=i_k<=j_k, i_k

1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 3, 0, 3, 1, 3, 2, 3, 1, 5, 3, 3, 2, 6, 4, 5, 3, 6, 6, 7, 2, 11, 5, 8, 6, 12, 7, 10, 8, 12, 11, 14, 8, 17, 11, 16, 13, 19, 13, 23, 15, 22, 17, 25, 18, 29, 24, 24, 23, 36, 25, 34, 25, 42, 34, 39, 30, 47, 40, 48, 37

Offset: 0

Author

Alois P. Heinz, May 06 2012

Keywords

• nonn

Examples

			a(0) = 1: 0 = the empty sum.
a(4) = 1: 4 = 2*2.
a(12) = 2: 12 = 2*6 = 3*4.
a(13) = 1: 13 = 2*2 + 3*3.
a(20) = 3: 20 = 2*2 + 4*4 = 2*10 = 4*5.
a(23) = 1: 23 = 2*4 + 3*5.
a(31) = 3: 31 = 2*5 + 3*7 = 2*3 + 5*5 = 2*2 + 3*9.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A066739, A182269, A182270, A211856, A211857, A212214, A212215, A212216, A212217, A212218.

Programs

Maple

with(numtheory):
b:= proc(n, m, i, j) option remember;
      `if`(n=0, 1, `if`(m<4, 0, b(n, m-1, i, j) +`if`(m>n, 0,
        add(b(n-m, m-1, min(i, k-1), min(j, m/k-1)), k=select(x->
         is(x>1 and x<=min(sqrt(m), i) and m<=j*x), divisors(m))))))
    end:
a:= n-> b(n$4):
seq(a(n), n=0..30);

Mathematica

b[n_, m_, i_, j_] := b[n, m, i, j] = If[n == 0, 1, If[m<4, 0, b[n, m-1, i, j] + If[m>n, 0, Sum[b[n-m, m-1, Min[i, k-1], Min[j, m/k-1]], {k, Select[Divisors[m], #>1 && # <= Min[Sqrt[m], i] && m <= j*# &]}]]]]; a[n_] := b[n, n, n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 09 2014, after Alois P. Heinz *)