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

1,10

Links

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

Example

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.

Maple

b:= proc(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))[]}))
    end:
a:= 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 *)

Crossrefs

The values v are in A213238.

Cf. A178830, A207852.

Sequence in context: A055172 A187445 A318362 * A029334 A275078 A286478

Adjacent sequences: A213234 A213235 A213236 * A213238 A213239 A213240

Keyword

nonn

Author

Alois P. Heinz, Jun 07 2012

Status

approved