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, 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

Matthew House, Table of n, a(n) for n = 1..10000

S. Alspaugh, Farmer Ted Goes 3D, Mathematics Magazine, Vol. 78, No. 3 (Jun., 2005), pp. 192-204.

M. DeLong, Undergraduate Mathematics Research at Taylor University [broken link]

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

Adjacent sequences: A048871 A048872 A048873 * A048875 A048876 A048877

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