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

1,2

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.

Links

Table of n, a(n) for n=1..49.

Formula

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

Example

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.

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);

Crossrefs

Cf. A230400 (subsequence), A066955.

Sequence in context: A154362 A226221 A072462 * A088827 A316900 A076057

Adjacent sequences: A369948 A369949 A369950 * A369952 A369953 A369954

Keyword

nonn,easy

Author

Felix Huber, Feb 12 2024

Status

approved