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A048874
Almost-cubes: numbers n such that n/s(n) >= k/s(k) for all k<n, where s(m) is the least surface area of a rectangular parallelepiped with integer side lengths and volume m.
1
1, 2, 3, 4, 6, 8, 12, 16, 18, 24, 27, 32, 36, 45, 48, 54, 60, 64, 72, 75, 80, 90, 96, 100, 112, 120, 125, 140, 144, 150, 168, 175, 180, 200, 210, 216, 240, 245, 252, 280, 288, 294, 320, 324, 336, 343, 378, 384, 392, 420, 432, 441, 448, 480, 486, 490, 504, 512
OFFSET
1,2
LINKS
S. Alspaugh, Farmer Ted Goes 3D, Mathematics Magazine, Vol. 78, No. 3 (Jun., 2005), pp. 192-204.
EXAMPLE
A rectangular parallelepiped with side lengths 1,2 and 3 has volume 6 and surface area 22. The ratio of volume to surface area is 6/22, which is greater than the ratio of volume to surface area for any rectangular parallelepiped with integer sides and volume < 6. Therefore 6 is an almost-cube.
MATHEMATICA
s[m_] := s[m] = First[Minimize[{a*b + b*c + c*a, a*b*c == m, 1 <= a <= b <= c}, {a, b, c}, Integers]]; almostCubeQ[ n_] := (r = For[k = 1, k < n, k++, If[n/s[n] < k/s[k], Return[False]] ] ; r =!= False); Reap[For[n = 1, n <= 512, n++, If[almostCubeQ[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 03 2012 *)
CROSSREFS
Cf. A033501.
Sequence in context: A326712 A191612 A362668 * A092824 A355578 A084094
KEYWORD
easy,nonn,nice
AUTHOR
Shawn Alspaugh (shalspau(AT)indiana.edu) and Matt DeLong (mtdelong(AT)tayloru.edu)
EXTENSIONS
Offset corrected by Matthew House, Aug 13 2024
STATUS
approved