A072368 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].

6, 54, 214, 594, 1334, 2614, 4645, 7676, 11992, 17912, 25791, 36021, 49028, 65269, 85247, 109493, 138575, 173094, 213694, 261048, 315863, 378888, 450907, 532730, 625213, 729244, 845748, 975679, 1120035, 1279848, 1456176, 1650123, 1862831, 2095469, 2349237

Offset: 1

Author

Wouter Meeussen, Jul 19 2002

Keywords

• nonn
• hard

Comments

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.

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

Examples

			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.
		

Links

• Martin Fuller, Table of n, a(n) for n = 1..80 (terms 1..50 from Rob Pratt)
• Martin Fuller, Illustration of initial terms
• Martin Fuller, Python program for this sequence
• Chai Wah Wu, On rearrangement inequalities for multiple sequences, arXiv:2002.10514 [math.CO], 2020.

Crossrefs

Cf. A070735.

Programs

Python

See Martin Fuller link

Extensions

Corrected and extended via integer linear programming by Rob Pratt, Jul 28 2023