A212217
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}, i_k*j_k
1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 3, 1, 3, 2, 5, 0, 7, 2, 8, 3, 10, 1, 15, 6, 14, 6, 21, 6, 28, 9, 26, 14, 38, 12, 50, 16, 47, 26, 70, 19, 82, 31, 87, 47, 111, 33, 141, 58, 143, 71, 182, 63, 228, 93, 231, 117, 289, 102, 364, 148, 354, 187, 462, 172, 537, 227
Offset: 0
Keywords
Examples
a(0) = 1: 0 = the empty sum. a(4) = 1: 4 = 2*2. a(6) = 1: 6 = 2*3. a(8) = 1: 8 = 2*4. a(9) = 1: 9 = 3*3. a(10) = 2: 10 = 2*2 + 2*3 = 2*5. a(12) = 3: 12 = 2*2 + 2*4 = 2*6 = 3*4. a(13) = 1: 13 = 2*2 + 3*3. a(14) = 3: 14 = 2*3 + 2*4 = 2*2 + 2*5 = 2*7. a(15) = 2: 15 = 2*3 + 3*3 = 3*5. a(16) = 5: 16 = 2*3 + 2*5 = 2*2 + 2*6 = 2*2 + 3*4 = 2*8 = 4*4. a(19) = 2: 19 = 2*2 + 2*3 + 3*3 = 2*2 + 3*5.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..200
Crossrefs
Programs
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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), min(j, m/k)), 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], Min[j, m/k]], {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, Jan 23 2017, after Alois P. Heinz *)