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A200743 Divide integers 1..n into two sets, minimizing the difference of their products. This sequence is the smaller product. 4
1, 1, 2, 4, 10, 24, 70, 192, 576, 1890, 6300, 21600, 78624, 294840, 1140480, 4561920, 18849600, 79968000, 348566400, 1559376000, 7147140000, 33522128640, 160745472000, 787652812800, 3938264064000, 20080974513600, 104348244639744, 552160113120000, 2973491173785600, 16286186592000000, 90678987245246400 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Michael S. Branicky, Table of n, a(n) for n = 1..35

EXAMPLE

For n=1, we put 1 in one set and the other is empty; with the standard convention for empty products, both products are 1.

For n=13, the central pair of divisors of n! are 78975 and 78848. Since neither is divisible by 10, these values cannot be obtained. The next pair of divisors are 79200 = 12*11*10*6*5*2*1 and 78624 = 13*9*8*7*4*3, so a(13) = 78624.

MAPLE

a:= proc(n) local l, ll, g, p, i; l:= [i$i=1..n]; ll:= [i!$i=1..n]; g:= proc(m, j, b) local mm, bb, k; if j=1 then m else mm:= m; bb:= b; for k to 2 while (mm<p) do if j=2 or k=2 or k=1 and ll[j-1]*mm>bb then bb:= max(bb, g(mm, j-1, bb)) fi; mm:= mm*l[j] od; bb fi end; Digits:= 700; p:= ceil(sqrt(ll[n])); g(1, nops(l), 1) end: seq(a(n), n=1..23);  # Alois P. Heinz, Nov 22 2011

MATHEMATICA

a[n_] := a[n] = Module[{s, t}, {s, t} = MinimalBy[{#, Complement[Range[n], #]}& /@ Subsets[Range[n]], Abs[Times @@ #[[1]] - Times @@ #[[2]]]&][[1]]; Min[Times @@ s, Times @@ t]];

Table[Print[n, " ", a[n]]; a[n], {n, 1, 25}] (* Jean-Fran├žois Alcover, Nov 03 2020 *)

PROG

(Python)

from itertools import combinations

def prod(l):

    t=1

    for x in l:

        t *= x

    return t

def a200743(n):

    nums = list(range(1, n+1))

    widths = combinations(nums, n//2)

    dimensions = [(prod(width), prod(x for x in nums if x not in width)) for width in widths]

    best = min(dimensions, key=lambda x:max(*x)-min(*x))

    return min(best)

# Christian Perfect, Feb 04 2015

CROSSREFS

Cf. A061055, A060776, A200744, A038667.

Sequence in context: A028506 A029893 A148089 * A060776 A061055 A148090

Adjacent sequences:  A200740 A200741 A200742 * A200744 A200745 A200746

KEYWORD

nonn

AUTHOR

Franklin T. Adams-Watters, Nov 21 2011

EXTENSIONS

a(24)-a(30) from Alois P. Heinz, Nov 22 2011

a(31) from Michael S. Branicky, May 21 2021

STATUS

approved

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Last modified July 27 21:21 EDT 2021. Contains 346316 sequences. (Running on oeis4.)