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%I #7 Sep 20 2024 06:43:41
%S 1,1,3,3,5,5,5,7,9,9,9,13,9,9,19,15,13,19,13,23,19,19,17,29,25,19,27,
%T 23,21,41,21,31,35,29,33,45,25,29,35,51,29,41,29,45,61,39,33,61,33,57,
%U 51,45,37,63,61,51,51,49,41,97,41,49,61,63,61,81,45,67,67
%N a(n) is the number of distinct integer-sided cuboids having the same surface as a cube with edge length n.
%C a(n) is the number of unordered solutions (x, y, z) to x*y + y*z + x*z = 3*n^2 in positive integers x and y.
%C Conjecture: All terms are odd.
%H Felix Huber, <a href="/A375785/b375785.txt">Table of n, a(n) for n = 1..10000</a>
%H Felix Huber, <a href="/A375785/a375785.txt">Maple programs</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Cuboid.html">Cuboid</a>
%e a(6) = 5 because exactly the 5 integer-sided cuboids (2, 2, 26), (2, 5, 14), (2, 6, 12), (3, 6, 10), (6, 6, 6) have the same surface as a cube with edge length 6: 2*(2*2 + 2*26 + 2*26) = 2*(2*5 + 5*14 + 2*14) = 2*(2*6 + 6*12 + 2*12) = 2*(3*6 + 6*10 + 3*10) = 2*(6*6 + 6*6 + 6*6) = 6*6^2.
%p See Huber link.
%Y Cf. A000578, A003167, A066955, A369951, A375580, A375786, A376074.
%K nonn
%O 1,3
%A _Felix Huber_, Sep 17 2024