

A102095


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


3



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, 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, 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
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OFFSET

1,2


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 for all n>1 a(n) <= n for all n. a(n) = n iff n is prime (a(1)=1).


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(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}


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][[1]], {k, 1, 84}]


CROSSREFS

Cf. A102096, A102097.
Sequence in context: A162325 A197862 A006530 * A109395 A145254 A163457
Adjacent sequences: A102092 A102093 A102094 * A102096 A102097 A102098


KEYWORD

nonn


AUTHOR

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


STATUS

approved



