A369951 Volumes of integer-sided cuboids in which either the surface area divides the volume or vice versa (assuming dimensionless unit of length).

1, 2, 4, 8, 16, 18, 27, 32, 36, 216, 250, 256, 288, 400, 432, 450, 486, 576, 882, 1728, 1800, 1944, 2000, 2048, 2304, 2744, 2916, 3200, 3456, 3528, 3600, 3888, 4608, 6144, 6174, 6750, 6912, 7056, 7200, 7350, 7776, 7986, 8000, 8100, 8232, 9000, 9216, 9600, 9800

Offset: 1

Author

Felix Huber, Feb 12 2024

Keywords

• nonn
• easy

Comments

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).

For 10 <= n <= 19 the surfaces and volumes are equal. This is sequence A230400.

For n >= 20 the volume divides the surface area.

Examples

			a(9) = 36, because V = 6*3*2 = 36 and S = 2*(6*3+3*2+6*2) = 72 and S/V = 2.
a(12) = 256, because V = 8*8*4 = 256 and S = 2*(8*8+8*4+8*4) = 256 and S=V.
a(20) = 1728, because V = 12*12*12 = 1728 and S = 6*12*12 = 864 and V/S = 2.
		

Crossrefs

Cf. A230400 (subsequence), A066955.

Programs

Maple

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);

Formula

For 10 <= n <= 19, a(n) = A230400(n - 9).