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%I #36 Feb 15 2024 12:40:18
%S 1,2,4,8,16,18,27,32,36,216,250,256,288,400,432,450,486,576,882,1728,
%T 1800,1944,2000,2048,2304,2744,2916,3200,3456,3528,3600,3888,4608,
%U 6144,6174,6750,6912,7056,7200,7350,7776,7986,8000,8100,8232,9000,9216,9600,9800
%N Volumes of integer-sided cuboids in which either the surface area divides the volume or vice versa (assuming dimensionless unit of length).
%C For n <= 9, the surface area divides the volume. The 9 triples with the edge lengths (u,v,w) are (1,1,1), (2,1,1), (2,2,1), (2,2,2), (4,4,1), (6,3,1), (3,3,3), (4,4,2), (6,3,2).
%C For 10 <= n <= 19 the surfaces and volumes are equal. This is sequence A230400.
%C For n >= 20 the volume divides the surface area.
%F For 10 <= n <= 19, a(n) = A230400(n - 9).
%e a(9) = 36, because V = 6*3*2 = 36 and S = 2*(6*3+3*2+6*2) = 72 and S/V = 2.
%e a(12) = 256, because V = 8*8*4 = 256 and S = 2*(8*8+8*4+8*4) = 256 and S=V.
%e a(20) = 1728, because V = 12*12*12 = 1728 and S = 6*12*12 = 864 and V/S = 2.
%p A369951 := proc(V) local a, b, c, k; for a from ceil(V^(1/3)) to V do if V/a = floor(V/a) then for b from ceil(sqrt(V/a)) to floor(V/a) do c := V/(a*b); if c = floor(c) then k := 2*(a*b + c*b + a*c)/(a*b*c); if k = floor(k) or 1/k = floor(1/k) then return V; end if; end if; end do; end if; end do; end proc; seq(A369951(V), V = 1 .. 10000);
%Y Cf. A230400 (subsequence), A066955.
%K nonn,easy
%O 1,2
%A _Felix Huber_, Feb 12 2024
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