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 A352813 Minimum difference |product(A) - product(B)| where A and B are a partition of {1,2,3,...,2*n} and |A| = |B| = n. 2
 0, 1, 2, 6, 18, 30, 576, 840, 24480, 93696, 800640, 7983360, 65318400, 2286926400, 13680979200, 797369149440, 16753029012720, 10176199188480, 159943859712000, 26453863460044800, 470500040794291200, 20720967220237197312, 61690805562507264000 (list; graph; refs; listen; history; text; internal format)
 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 A064842 Adjacent sequences:  A352810 A352811 A352812 * A352814 A352815 A352816 KEYWORD nonn AUTHOR Peter J. Taylor, Apr 04 2022 STATUS approved

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Last modified August 9 18:55 EDT 2022. Contains 356026 sequences. (Running on oeis4.)