OFFSET
0,4
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
LINKS
KöMaL-Mathematical and Physical Journal for Secondary Schools, Problems in Mathematics, September 2015.
EXAMPLE
For n = 4, the four possible values of |product(A) - product(B)| are 2, 5, 10, and 23.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Jerrold Grossman, Oct 13 2015
EXTENSIONS
a(21)-a(27) from Giovanni Resta, Oct 17 2015
a(28)-a(38) from Alois P. Heinz, Oct 17 2015
STATUS
approved