login
Minimal total volume of n bricks with integer sides, all sides being different. Lowest value of sum of products of triples p*q*r chosen from [1,3n].
3

%I #44 Aug 05 2023 21:29:38

%S 6,54,214,594,1334,2614,4645,7676,11992,17912,25791,36021,49028,65269,

%T 85247,109493,138575,173094,213694,261048,315863,378888,450907,532730,

%U 625213,729244,845748,975679,1120035,1279848,1456176,1650123,1862831,2095469,2349237

%N Minimal total volume of n bricks with integer sides, all sides being different. Lowest value of sum of products of triples p*q*r chosen from [1,3n].

%C For n=19, the smallest integer from each triple does not belong to range [1,19]. Triplicating the sets of triples, shifting each triple to the left, generates permutations as in A070735, but not provably minimal ones.

%C a(n) >= ceiling(n*(3n!)^(1/n)) with the inequality tight for 1 <= n <= 3. - _Chai Wah Wu_, Mar 05 2020

%H Martin Fuller, <a href="/A072368/b072368.txt">Table of n, a(n) for n = 1..80</a> (terms 1..50 from Rob Pratt)

%H Martin Fuller, <a href="/A072368/a072368_3.txt">Illustration of initial terms</a>

%H Martin Fuller, <a href="/A072368/a072368_4.txt">Python program for this sequence</a>

%H Chai Wah Wu, <a href="https://arxiv.org/abs/2002.10514">On rearrangement inequalities for multiple sequences</a>, arXiv:2002.10514 [math.CO], 2020.

%e a(7)=4645 because (1*20*21)+(2*18*19)+(3*15*16)+(4*13*14)+(5*8*17)+(6*10*12)+(7*9*11)=4645 is the smallest value attainable.

%o (Python) See Martin Fuller link

%Y Cf. A070735.

%K nonn,hard

%O 1,1

%A _Wouter Meeussen_, Jul 19 2002

%E Corrected and extended via integer linear programming by _Rob Pratt_, Jul 28 2023