A213238 Triangle T(n,k) in which n-th row lists in increasing order the distinct values v satisfying v = sum of elements in S = product of elements in P for a partition of {1,...,n} into two sets S and P.
1, 3, 8, 12, 18, 24, 32, 40, 42, 50, 60, 64, 72, 84, 88, 90, 98, 99, 105, 112, 120, 128, 130, 135, 144, 162, 168, 180, 182, 192, 200, 208, 210, 220, 231, 242, 252, 264, 266, 272, 280, 288, 294, 300, 312, 315, 320, 324, 330, 338, 340, 360, 364, 378, 392, 400
Offset: 1
Examples
For n=1 v=1 satisfies the condition with S={1}, P={} => row 1 = [1].
For n=2 no v can be found => row 2 is empty: [].
For n=3 there is one solution: S={1,2}, P={3}, v=3 => row 3 = [3].
For n=10 we have three partitions of {1,2,...,10} into S and P satisfying v = Sum_{i:S} i = Product_{k:P} k but there are only two distinct values v: S={2,3,5,6,7,8,9}, P={1,4,10}, v=40; S={4,5,6,8,9,10}, P={1,2,3,7}, v=42; S={1,2,3,4,5,8,9,10}, P={6,7}, v=42 => row 10 = [40, 42].
Triangle T begins:
1;
;
3;
;
8;
12;
18;
24;
32;
40, 42;
50;
60, 64;
72;
84, 88, 90;
...
Links
- Alois P. Heinz, Rows n = 1..798, flattened
Programs
-
Maple
b:= proc(n, s, p) `if`(s=p, {s}, `if`(n<1, {}, {b(n-1, s, p)[], `if`(s-n -
Mathematica
b[n_, s_, p_] := If[s == p, {s}, If[n < 1, {}, {b[n-1, s, p], If[s-n < p*n, {}, b[n-1, s-n, p*n]]} // Union]]; T[n_] := Sort[b[n, n(n+1)/2, 1] // Flatten] // Union; Array[T, 30] // Flatten (* Jean-François Alcover, Feb 19 2021, after Alois P. Heinz *)
Formula
T(n,1) = floor((n-1)^2/2) = A007590(n-1) for n>=5.