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%I #37 Apr 07 2022 10:41:24
%S 1,1,1,2,4,8,13,26,44,76,119,238,324,648,1008,1492,2116,4232,5680,
%T 11360,15272,21872,33536,67072,83168,121376,185496,249072,328416,
%U 656832,790656,1581312,1980192,2758624,4193040,5555616,6532896,13065792,19845216
%N Number of distinct values of |product(A) - product(B)| where A and B are a partition of {1,2,...,n}.
%C 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).
%C Compare to A038667, which provided the smallest value of |product(A) - product(B)|.
%C Also the number of distinct values <= sqrt(n!) of element products of subsets of [n]. - _Alois P. Heinz_, Oct 17 2015
%H KöMaL-Mathematical and Physical Journal for Secondary Schools, <a href="http://www.komal.hu/verseny/feladat.cgi?a=honap&h=201509&t=mat&l=en">Problems in Mathematics</a>, September 2015.
%e For n = 4, the four possible values of |product(A) - product(B)| are 2, 5, 10, and 23.
%p b:= proc(n) option remember; local f, g, h;
%p if n<2 then {1}
%p else f, g, h:= n!, y-> `if`(y^2<=f, y, NULL), (n-1)!;
%p map(x-> {x, g(x*n), g(h/x)}[], b(n-1))
%p fi
%p end:
%p a:= n-> nops(b(n)):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Oct 17 2015
%t 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 *)
%o (Python)
%o from math import prod, factorial
%o from itertools import combinations
%o def A263292(n):
%o m = factorial(n)
%o 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
%Y Cf. A038667.
%K nonn
%O 0,4
%A _Jerrold Grossman_, Oct 13 2015
%E a(21)-a(27) from _Giovanni Resta_, Oct 17 2015
%E a(28)-a(38) from _Alois P. Heinz_, Oct 17 2015
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