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Square of the minimal space diagonal of an integral cuboid having volume n.
1

%I #7 Mar 05 2023 12:40:00

%S 3,6,11,9,27,14,51,12,19,30,123,17,171,54,35,24,291,22,363,33,59,126,

%T 531,29,51,174,27,57,843,38,963,36,131,294,75,34,1371,366,179,45,1683,

%U 62,1851,129,43,534,2211,41,99,54,299,177,2811,54,147,69,371,846,3483

%N Square of the minimal space diagonal of an integral cuboid having volume n.

%C 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.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Cuboid.html">Cuboid</a>.

%e 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.

%t 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}]

%Y Cf. A102890.

%K nonn

%O 1,1

%A Joseph Biberstine (jrbibers(AT)indiana.edu), Jan 16 2005