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A102889
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Square of the minimal space diagonal of an integral cuboid having volume n.
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1
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3, 6, 11, 9, 27, 14, 51, 12, 19, 30, 123, 17, 171, 54, 35, 24, 291, 22, 363, 33, 59, 126, 531, 29, 51, 174, 27, 57, 843, 38, 963, 36, 131, 294, 75, 34, 1371, 366, 179, 45, 1683, 62, 1851, 129, 43, 534, 2211, 41, 99, 54, 299, 177, 2811, 54, 147, 69, 371, 846, 3483
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OFFSET
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1,1
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COMMENTS
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a(n) = A102095(n)^2 + A102096(n)^2 + A102097(n)^2. Ceiling(3*n^(2/3)) <= a(n) <= 2 + n^2 a(n) = 3*n^(2/3) iff n is a cube. a(n) = 2 + n^2 iff n is prime.
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LINKS
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Eric Weisstein's World of Mathematics, Cuboid.
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EXAMPLE
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a(4)=9 because the minimal-space-diagonal integral cuboid of volume 4 has dimensions 2 by 2 by 1 and 2^2 + 2^2 + 1^2 = 9.
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MATHEMATICA
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Clear[surfarea, sumprint, temp, 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]]) surfarea[n_] := (Clear[temp]; temp = fac[n]; 2*temp[[1]]^2 + 2*temp[[2]]^2 + 2*temp[[3]]^2) Table[surfarea[n]/2, {n, 1, 85}]
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CROSSREFS
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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), Jan 16 2005
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STATUS
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approved
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