A102096 Second greatest (or second least) edge length of a cuboid having integer edge lengths, volume n and minimal surface area under those restrictions.

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 3, 5, 2, 3, 2, 1, 3, 1, 4, 3, 2, 5, 3, 1, 2, 3, 4, 1, 3, 1, 2, 3, 2, 1, 4, 7, 5, 3, 2, 1, 3, 5, 4, 3, 2, 1, 4, 1, 2, 3, 4, 5, 3, 1, 2, 3, 5, 1, 4, 1, 2, 5, 2, 7, 3, 1, 4, 3, 2, 1, 4

Offset

1,4

Comments

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 iff n is prime (a(1)=1). a(n) <= n^(1/2) for all n.

Links

Table of n, a(n) for n=1..84.

Eric Weisstein's World of Mathematics, Cuboid

Eric Weisstein's World of Mathematics, Sample Variance

Wikipedia, Nonlinear Programming

Example

a(10) = 2 because the cuboid of integer edge lengths, volume = 10 and minimal possible surface area under those restrictions has edge lengths {5,2,1}

Mathematica

Clear[fac, faclist, red, bool, n, a, b, c, i, ai, bi, ci]
red[n_] := Reduce[{a*b*c == n, a >= b >= c > 0}, {a, b, c}, Integers];
faclist[n_] := (
If[PrimeQ[n] || n == 1, Return[{n + 1 + 1, {n, 1, 1}}]; Abort[]];
bool = red[n];
Reap[For[i = 1, i <= Length[bool], i++,
ai = bool[[i]][[1]][[2]];
bi = bool[[i]][[2]][[2]];
ci = bool[[i]][[3]][[2]];
Sow[{ai + bi + ci, {ai, bi, ci}}]]][[2]][[1]])
fac[n_] := (
If[PrimeQ[n] || n == 1, Return[{n, 1, 1}]; Abort[]];
faclist[n][[1]][[2]])
Table[fac[k][[2]], {k, 1, 84}]

Crossrefs

Cf. A102095, A102097.

Sequence in context: A107286 A339791 A087039 * A322813 A366189 A137866

Adjacent sequences: A102093 A102094 A102095 * A102097 A102098 A102099

Keyword

nonn

Author

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

Status

approved