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<i_{k+1}, j_k<j_{k+1}.

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

0,13

Links

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A265752 A125090 A294880 * A073266 A125692 A128258

Adjacent sequences: A212216 A212217 A212218 * A212220 A212221 A212222

Keyword

nonn

Author

Alois P. Heinz, May 06 2012

Status

approved