A375785 -id:A375785 - cp's OEIS frontend

A375786 a(n) is the minimum volume of an integer-sided cuboid having the same surface as a cube with edge length n.

1, 8, 13, 36, 37, 104, 73, 188, 121, 252, 181, 428, 253, 540, 337, 764, 433, 828, 541, 1196, 661, 1448, 793, 1476, 937, 2024, 1093, 2160, 1261, 2592, 1441, 2628, 1633, 3464, 1837, 3884, 2053, 3708, 2281, 4796, 2521, 5148, 2773, 5616, 3037, 5436, 3313, 6660, 3601

Offset: 1

Felix Huber, Sep 17 2024

nonn

Conjecture: From the integer-sided cuboids with same surface 6*n^2 always the one with the smallest edge length has the minimum volume. If there are several integer-sided cuboids having the smallest edge length, then the one with the smallest second smallest edge length has the minimum volume (checked up to a(1000)).

The maximum volume is always A000578(n) = n^3.

			a(6) = 104: because from the five integer-sided cuboids (2, 2, 26), (2, 5, 14), (2, 6, 12), (3, 6, 10), (6, 6, 6) having the same surface as a cube with edge length 6 (see example in A375785) has (2, 2, 26) with 2*2*26 = 104 the smallest volume.

Felix Huber, Table of n, a(n) for n = 1..10000
Felix Huber, Maple programs
Eric Weisstein's World of Mathematics, Cuboid

Cf. A000578, A369951, A375580, A375785.

Maple
```
See Huber link.
```

A376074 a(n) is the number of distinct right circular cones with integer radius and height having the same volume as a sphere with radius n.

Original entry on oeis.org

2, 3, 4, 5, 4, 6, 4, 6, 8, 6, 4, 10, 4, 6, 8, 8, 4, 12, 4, 10, 8, 6, 4, 12, 8, 6, 10, 10, 4, 12, 4, 9, 8, 6, 8, 20, 4, 6, 8, 12, 4, 12, 4, 10, 16, 6, 4, 16, 8, 12, 8, 10, 4, 15, 8, 12, 8, 6, 4, 20, 4, 6, 16, 11, 8, 12, 4, 10, 8, 12, 4, 24, 4, 6, 16, 10, 8, 12, 4

Offset: 1

Felix Huber, Sep 20 2024

nonn

a(n) is also the number of solutions to x^2*y = 4*n^3 in positive integers x and y.

			a(3) = 4 counts the following right circular cones (r, h): (1, 108), (2, 27), (3, 12), (6, 3). These 4 cones have the same volume as a sphere with radius 3: (1/3)*Pi*1^2*108 = (1/3)*Pi*2^2*27 = (1/3)*Pi*3^2*12 = (1/3)*Pi*6^2*3 = (4/3)*Pi*3^3 = 36*Pi.

Felix Huber, Table of n, a(n) for n = 1..10000
Felix Huber, Maple programs
Eric Weisstein's World of Mathematics, Cone
Eric Weisstein's World of Mathematics, Sphere

Cf. A375576, A375580, A375785.

Maple
```
See Huber link.
```

A377133 Triangle read by rows: T(n,k) is the maximum volume of an integer-sided box that can be made from a piece of paper of size n X k by cutting away identical squares at each corner and folding up the sides, n >= 3, 3 <= k <= n.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 4, 8, 12, 16, 5, 10, 15, 20, 25, 6, 12, 18, 24, 30, 36, 7, 14, 21, 28, 35, 42, 50, 8, 16, 24, 32, 40, 48, 60, 72, 9, 18, 27, 36, 45, 56, 70, 84, 98, 10, 20, 30, 40, 50, 64, 80, 96, 112, 128, 11, 22, 33, 44, 55, 72, 90, 108, 126, 144, 162, 12, 24

Offset: 3

Felix Huber, Oct 25 2024

For a sketch see linked illustration "Box made from nXk-paper".

The first few rows follow (n-2) * (k-2), so the initial terms are the same as in A075362. The first difference is at T(9,9) = 50 which is greater than 7 * 7.

			Triangle T(n,k) begins:
   n\k   3     4     5     6     7     8     9    10    11    12    13 ...
   3     1
   4     2     4
   5     3     6     9
   6     4     8    12    16
   7     5    10    15    20    25
   8     6    12    18    24    30    36
   9     7    14    21    28    35    42    50
  10     8    16    24    32    40    48    60    72
  11     9    18    27    36    45    56    70    84    98
  12    10    20    30    40    50    64    80    96   112   128
  13    11    22    33    44    55    72    90   108   126   144   162

Felix Huber, Rows n = 3..142 of triangle, flattened
Felix Huber, Box made from nXk-paper

Cf. A075362, A355880, A375580, A375785, A375785.

A377113:=proc(n,k)
   local a,x,V;
   a:=0;
   for x to (k-1)/2 do
      V:=x*(n-2*x)*(k-2*x);
      if V>a then
         a:=V
      fi
   od;
   return a
end proc;
seq(seq(A377113(n,k),k=3..n),n=3..14);

T(n,k) = (n-2*x)*(k-2*x)*x with x = round((n+k-(sqrt(n^2+k^2-n*k)))/6).

cp's OEIS Frontend

A375786 a(n) is the minimum volume of an integer-sided cuboid having the same surface as a cube with edge length n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A376074 a(n) is the number of distinct right circular cones with integer radius and height having the same volume as a sphere with radius n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A377133 Triangle read by rows: T(n,k) is the maximum volume of an integer-sided box that can be made from a piece of paper of size n X k by cutting away identical squares at each corner and folding up the sides, n >= 3, 3 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula