A102097 Least edge length of a cuboid having integer edge lengths, volume n and minimal surface area under those restrictions.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 3, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 3, 2, 1, 2, 1, 4, 3, 1, 1, 3

Offset

1,8

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 or n is semiprime (a(1)=1). a(n) <= n^(1/3) for all n.

Links

Hans Havermann, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Cuboid

Eric Weisstein's World of Mathematics, Sample Variance

Wikipedia, Nonlinear programming

Example

a(10) = 1 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][[3]], {k, 1, 84}]

Crossrefs

Cf. A102095, A102096.

Sequence in context: A293234 A347441 A360617 * A361566 A354991 A354990

Adjacent sequences: A102094 A102095 A102096 * A102098 A102099 A102100

Keyword

nonn

Author

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

Status

approved