A352813 Minimum difference |product(A) - product(B)| where A and B are a partition of {1,2,3,...,2*n} and |A| = |B| = n.

0, 1, 2, 6, 18, 30, 576, 840, 24480, 93696, 800640, 7983360, 65318400, 2286926400, 13680979200, 797369149440, 16753029012720, 10176199188480, 159943859712000, 26453863460044800, 470500040794291200, 20720967220237197312, 61690805562507264000

Offset

0,3

Comments

a(n) >= A038667(2*n).

Conjecture: a(n) = A038667(2*n) for all n. It is verified for n<=70. - Max Alekseyev, Jun 18 2022

Bernardo Recamán Santos proposes that this should be called Luciana's sequence for the student whose question prompted its investigation. (See MathOverflow link below.)

Links

Max Alekseyev, Table of n, a(n) for n = 0..70

Gordon Hamilton, Thirsty Fractions, MathPickle, 2013, for elementary and middle school teachers.

MathOverflow discussion Splitting the integers from 1 to 2n into two sets with products as close as possible.

Example

For n = 4, the partition A = {1,5,6,7} and B = {2,3,4,8} is optimal, giving difference 1*5*6*7 - 2*3*4*8 = 18.
Rob Pratt computed the optimal solutions for n <= 10:
[ n]    a(n)                   partitions of 2n
------------------------------------------------------------------
[ 1]       1                         2 | 1
[ 2]       2                       2,3 | 1,4
[ 3]       6                     1,5,6 | 2,3,4
[ 4]      18                   1,5,6,7 | 2,3,4,8
[ 5]      30                2,3,4,8,10 | 1,5,6,7,9
[ 6]     576              1,4,7,8,9,11 | 2,3,5,6,10,12
[ 7]     840           2,4,5,6,8,11,14 | 1,3,7,9,10,12,13
[ 8]   24480        1,5,6,7,8,13,14,15 | 2,3,4,9,10,11,12,16
[ 9]   93696     2,3,6,8,9,11,12,13,18 | 1,4,5,7,10,14,15,16,17
[10]  800640  2,3,4,8,9,11,12,18,19,20 | 1,5,6,7,10,13,14,15,16,17

Prog

(Sage)
def A352813(n):
    return min(abs(prod(A)-prod(B)) for (A, B) in SetPartitions((1..2*n), [n, n]))
[A352813(n) for n in (1..10)] # Freddy Barrera, Apr 05 2022
(Python)
from math import prod, factorial
from itertools import combinations
def A352813(n):
    m = factorial(2*n)
    return 0 if n == 0 else min(abs((p:=prod(d))-m//p) for d in combinations(range(2, 2*n+1), n-1)) # Chai Wah Wu, Apr 06 2022

Crossrefs

Cf. A061057, A038667.

Sequence in context: A277200 A277324 A034881 * A146345 A328633 A368566

Adjacent sequences: A352810 A352811 A352812 * A352814 A352815 A352816

Keyword

nonn

Author

Peter J. Taylor, Apr 04 2022

Status

approved