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A263292
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Number of distinct values of |product(A) - product(B)| where A and B are a partition of {1,2,...,n}.
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4
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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
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OFFSET
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0,4
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COMMENTS
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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
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LINKS
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Table of n, a(n) for n=0..38.
KöMaL-Mathematical and Physical Journal for Secondary Schools, Problems in Mathematics, September 2015.
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EXAMPLE
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For n = 4, the four possible values of |product(A) - product(B)| are 2, 5, 10, and 23.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A038667.
Sequence in context: A043816 A048328 A094767 * A026643 A288925 A018285
Adjacent sequences: A263289 A263290 A263291 * A263293 A263294 A263295
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KEYWORD
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nonn
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AUTHOR
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Jerrold Grossman, Oct 13 2015
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EXTENSIONS
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a(21)-a(27) from Giovanni Resta, Oct 17 2015
a(28)-a(38) from Alois P. Heinz, Oct 17 2015
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STATUS
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approved
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