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A102097
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Least edge length of a cuboid having integer edge lengths, volume n and minimal surface area under those restrictions.
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3
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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
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OFFSET
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1,8
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=1..84.
Eric Weisstein's World of Mathematics, "Cuboid."
Eric Weisstein's World of Mathematics, "Sample Variance."
Various (Wikipedia), "Nonlinear Programming."
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EXAMPLE
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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}
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MATHEMATICA
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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}]
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CROSSREFS
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Cf. A102095, A102096.
Sequence in context: A161099 A118663 A205119 * A050330 A205788 A214054
Adjacent sequences: A102094 A102095 A102096 * A102098 A102099 A102100
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KEYWORD
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nonn
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AUTHOR
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Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 29 2004
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STATUS
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approved
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