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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.
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%I #25 Aug 14 2024 03:09:46

%S 1,2,3,4,6,8,12,16,18,24,27,32,36,45,48,54,60,64,72,75,80,90,96,100,

%T 112,120,125,140,144,150,168,175,180,200,210,216,240,245,252,280,288,

%U 294,320,324,336,343,378,384,392,420,432,441,448,480,486,490,504,512

%N 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.

%H Matthew House, <a href="/A048874/b048874.txt">Table of n, a(n) for n = 1..10000</a>

%H S. Alspaugh, <a href="https://www.jstor.org/stable/30044156">Farmer Ted Goes 3D</a>, Mathematics Magazine, Vol. 78, No. 3 (Jun., 2005), pp. 192-204.

%H M. DeLong, <a href="http://www.css.tayloru.edu/~mdelong/undergrad.html">Undergraduate Mathematics Research at Taylor University</a> [broken link]

%e 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.

%t 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 *)

%Y Cf. A033501.

%K easy,nonn,nice

%O 1,2

%A Shawn Alspaugh (shalspau(AT)indiana.edu) and Matt DeLong (mtdelong(AT)tayloru.edu)

%E Offset corrected by _Matthew House_, Aug 13 2024