A213237 Number of distinct values v satisfying v = sum of elements in S = product of elements in P for any partition of {1,...,n} into two sets S and P.

1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 3, 2, 3, 1, 2, 3, 3, 2, 2, 4, 3, 5, 3, 2, 3, 3, 4, 4, 5, 1, 3, 2, 4, 4, 6, 3, 3, 2, 3, 4, 9, 3, 4, 9, 4, 3, 5, 4, 4, 4, 6, 6, 5, 5, 4, 7, 4, 8, 6, 4, 7, 3, 6, 5, 3, 4, 6, 5, 4, 6, 6, 5, 7, 4, 6, 9, 7, 6, 6, 8, 4, 7, 5

Offset: 1

Author

Alois P. Heinz, Jun 07 2012

Keywords

• nonn

Examples

			a(1) = 1: S={1}, P={}, v=1.
a(2) = 0: no partition of {1,2} satisfies the condition.
a(3) = 1: S={1,2}, P={3}, v=3.
a(10) = 2: three partitions of {1,2,...,10} into S and P satisfy 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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

The values v are in A213238.

Cf. A178830, A207852.

Programs

Maple

b:= proc(n, s, p)
      `if`(s=p, {s}, `if`(n<1, {}, {b(n-1, s, p)[],
      `if`(s-n nops(b(n, n*(n+1)/2, 1)):
seq(a(n), n=1..100);

Mathematica

b[n_, s_, p_] := b[n, s, p] = If[s == p, {s}, If[n < 1, {}, Union[b[n - 1, s, p], If[s - n < p n, {}, b[n - 1, s - n, p n]]]]];
a[n_] := Length[b[n, n(n+1)/2, 1]];
Array[a, 100] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)