A263292 Number of distinct values of |product(A) - product(B)| where A and B are a partition of {1,2,...,n}.

1, 1, 1, 2, 4, 8, 13, 26, 44, 76, 119, 238, 324, 648, 1008, 1492, 2116, 4232, 5680, 11360, 15272, 21872, 33536, 67072, 83168, 121376, 185496, 249072, 328416, 656832, 790656, 1581312, 1980192, 2758624, 4193040, 5555616, 6532896, 13065792, 19845216

Offset: 0

Author

Jerrold Grossman, Oct 13 2015

Keywords

• nonn

Comments

The problem of showing that no number k is equal to |product(A)-product(B)| for infinitely many different values of n appears in a Hungarian journal for high school students in math and physics (see KöMaL link).

Compare to A038667, which provided the smallest value of |product(A) - product(B)|.

Also the number of distinct values <= sqrt(n!) of element products of subsets of [n]. - Alois P. Heinz, Oct 17 2015

Examples

			For n = 4, the four possible values of |product(A) - product(B)| are 2, 5, 10, and 23.
		

Links

• KöMaL-Mathematical and Physical Journal for Secondary Schools, Problems in Mathematics, September 2015.

Crossrefs

Cf. A038667.

Programs

Maple

b:= proc(n) option remember; local f, g, h;
      if n<2 then {1}
    else f, g, h:= n!, y-> `if`(y^2<=f, y, NULL), (n-1)!;
         map(x-> {x, g(x*n), g(h/x)}[], b(n-1))
      fi
    end:
a:= n-> nops(b(n)):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 17 2015

Mathematica

a[n_] := Block[{v = Times @@@ Subsets[ Range[2, n], Floor[n/2]]}, Length@ Union@ Abs[v - n!/v]]; Array[a, 20] (* Giovanni Resta, Oct 17 2015 *)

Python

from math import prod, factorial
from itertools import combinations
def A263292(n):
    m = factorial(n)
    return 1 if n == 0 else len(set(abs((p:=prod(d))-m//p) for l in range(n,n//2,-1) for d in combinations(range(1,n+1),l))) # Chai Wah Wu, Apr 07 2022

Extensions

a(21)-a(27) from Giovanni Resta, Oct 17 2015

a(28)-a(38) from Alois P. Heinz, Oct 17 2015