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A263292 Number of distinct values of |product(A) - product(B)| where A and B are a partition of {1,2,...,n}. 2
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..38.

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 *)

CROSSREFS

Cf. A038667.

Sequence in context: A043816 A048328 A094767 * A026643 A288925 A018285

Adjacent sequences:  A263289 A263290 A263291 * A263293 A263294 A263295

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

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Last modified August 7 15:08 EDT 2020. Contains 336276 sequences. (Running on oeis4.)