%I A102889
%S A102889 3,6,11,9,27,14,51,12,19,30,123,17,171,54,35,24,291,22,363,33,59,126,
%T A102889 531,29,51,174,27,57,843,38,963,36,131,294,75,34,1371,366,179,45,1683,
%U A102889 62,1851,129,43,534,2211,41,99,54,299,177,2811,54,147,69,371,846,3483
%N A102889 Square of the minimal space diagonal of an integral cuboid having volume 
               n.
%C A102889 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 A102889 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Cuboid.html">"Cuboid."</a>
%e A102889 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 A102889 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 A102889 Cf. A102890.
%Y A102889 Sequence in context: A117128 A006509 A144562 this_sequence A028744 A028775 
               A022155
%Y A102889 Adjacent sequences: A102886 A102887 A102888 this_sequence A102890 A102891 
               A102892
%K A102889 nonn
%O A102889 1,1
%A A102889 Joseph Biberstine (jrbibers(AT)indiana.edu), Jan 16 2005