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A102095 Greatest edge length of a cuboid having integer edge lengths, volume n and minimal surface area under those restrictions. 3

%I

%S 1,2,3,2,5,3,7,2,3,5,11,3,13,7,5,4,17,3,19,5,7,11,23,4,5,13,3,7,29,5,

%T 31,4,11,17,7,4,37,19,13,5,41,7,43,11,5,23,47,4,7,5,17,13,53,6,11,7,

%U 19,29,59,5,61,31,7,4,13,11,67,17,23,7,71,6,73,37,5,19,11,13,79,5,9,41,83,7

%N Greatest edge length of a cuboid having integer edge lengths, volume n and minimal surface area under those restrictions.

%C Finding a(n) given n is a fundamental problem from integer nonlinear programming, equivalent to minimizing the sum a+b+c when a*b*c=n and a,b,c are integers. a(n) is not strictly prime. a(n) > 1 for all n>1 a(n) <= n for all n. a(n) = n iff n is prime (a(1)=1).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Cuboid.html">"Cuboid."</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SampleVariance.html">"Sample Variance."</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Nonlinear_programming">"Nonlinear Programming."</a>

%e a(16) = 4 because the cuboid of integer edge lengths, volume = 16 and minimal possible surface area under those restrictions has edge lengths {4,2,2}

%t Clear[fac, faclist, red, bool, n, a, b, c, i, ai, bi, ci]

%t red[n_] := Reduce[{a*b*c == n, a >= b >= c > 0}, {a, b, c}, Integers];

%t faclist[n_] := (

%t If[PrimeQ[n] || n == 1, Return[{n + 1 + 1, {n, 1, 1}}]; Abort[]];

%t bool = red[n];

%t Reap[For[i = 1, i <= Length[bool], i++,

%t ai = bool[[i]][[1]][[2]];

%t bi = bool[[i]][[2]][[2]];

%t ci = bool[[i]][[3]][[2]];

%t Sow[{ai + bi + ci, {ai, bi, ci}}]]][[2]][[1]])

%t fac[n_] := (

%t If[PrimeQ[n] || n == 1, Return[{n, 1, 1}]; Abort[]];

%t faclist[n][[1]][[2]])

%t Table[fac[k][[1]], {k, 1, 84}]

%Y Cf. A102096, A102097.

%K nonn

%O 1,2

%A Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 29 2004

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