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A369951
Volumes of integer-sided cuboids in which either the surface area divides the volume or vice versa (assuming dimensionless unit of length).
4
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.
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
KEYWORD
nonn,easy
AUTHOR
Felix Huber, Feb 12 2024
STATUS
approved