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
KEYWORD
nonn
AUTHOR
Peter J. Taylor, Apr 04 2022
STATUS
approved