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A102096
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Second greatest (or second 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, 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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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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.
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LINKS
| Eric Weisstein's World of Mathematics, "Cuboid."
Eric Weisstein's World of Mathematics, "Sample Variance."
Wikipedia, "Nonlinear Programming."
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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}
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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}]
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CROSSREFS
| Cf. A102095, A102097.
Sequence in context: A082061 A107286 A087039 * A137866 A072203 A124044
Adjacent sequences: A102093 A102094 A102095 * A102097 A102098 A102099
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KEYWORD
| nonn
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AUTHOR
| Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 29 2004
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