A375580 -id:A375580 - cp's OEIS frontend

A375576 a(n) is the number of partitions n = x + y + z of positive integers such that xyz is a perfect square.

0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 4, 1, 2, 3, 2, 2, 4, 2, 4, 3, 4, 2, 5, 4, 2, 6, 5, 2, 8, 4, 8, 4, 4, 5, 10, 5, 3, 8, 7, 6, 12, 5, 6, 7, 6, 7, 11, 5, 6, 8, 12, 6, 11, 8, 11, 11, 6, 3, 22, 6, 12, 12, 8, 9, 13, 12, 7, 14, 14, 6, 18, 7, 7, 18, 13, 14, 13, 7, 19, 10

Offset: 0

Felix Huber, Aug 19 2024

nonn

a(n) is also the number of distinct integer-sided cuboids with total edge length 4*n whose unit cubes can be grouped to a square cuboid with height 1.

			a(24) = 4 because the four partitions [2, 4, 18], [3, 9, 12], [4, 4, 16], [4, 10, 10] satisfy the conditions: 2 + 4 + 18 = 24 and 2*4*18 = 12^2, 3 + 9 + 12 = 24 and 3*9*12 = 18^2, 4 + 4 + 16 = 24 and 4*4*16 = 16^2, 4 + 10 + 10 = 24 and 4*10*10 = 20^2.
See also linked Maple code.

Felix Huber, Table of n, a(n) for n = 0..1000
Felix Huber, Maple Codes

Cf. A338939, A375512, A375580.

Maple
```
See Huber link.
```

from sympy.ntheory.primetest import is_square
def A375567(n): return sum(1 for x in range(n//3) for y in range(x,n-x-1>>1) if is_square((n-x-y-2)*(x+1)*(y+1))) # Chai Wah Wu, Aug 22 2024

A375785 a(n) is the number of distinct integer-sided cuboids having the same surface as a cube with edge length n.

Original entry on oeis.org

1, 1, 3, 3, 5, 5, 5, 7, 9, 9, 9, 13, 9, 9, 19, 15, 13, 19, 13, 23, 19, 19, 17, 29, 25, 19, 27, 23, 21, 41, 21, 31, 35, 29, 33, 45, 25, 29, 35, 51, 29, 41, 29, 45, 61, 39, 33, 61, 33, 57, 51, 45, 37, 63, 61, 51, 51, 49, 41, 97, 41, 49, 61, 63, 61, 81, 45, 67, 67

Offset: 1

Felix Huber, Sep 17 2024

nonn

a(n) is the number of unordered solutions (x, y, z) to x*y + y*z + x*z = 3*n^2 in positive integers x and y.

Conjecture: All terms are odd.

			a(6) = 5 because exactly the 5 integer-sided cuboids (2, 2, 26), (2, 5, 14), (2, 6, 12), (3, 6, 10), (6, 6, 6) have the same surface as a cube with edge length 6: 2*(2*2 + 2*26 + 2*26) = 2*(2*5 + 5*14 + 2*14) = 2*(2*6 + 6*12 + 2*12) = 2*(3*6 + 6*10 + 3*10) = 2*(6*6 + 6*6 + 6*6) = 6*6^2.

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

Cf. A000578, A003167, A066955, A369951, A375580, A375786, A376074.

Maple
```
See Huber link.
```

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

Original entry on oeis.org

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.
```

A375618 a(n) is the least positive integer k such that there are n partitions k = x + y + z of positive integers such that x * y * z is a perfect cube or -1 if no such positive integer exists.

Original entry on oeis.org

1, 3, 20, 21, 57, 94, 133, 219, 217, 273, 453, 434, 551, 589, 399, 791, 665, 893, 1321, 779, 1330, 1387, 1519, 1749, 1786, 2033, 1767, 2527, 2793, 1995, 4066, 3325, 4389, 5548, 4557, 3895, 4123, 5187, 5890, 5529, 5453, 8075, 6213, 7980, 7581, 7790, 11275, 8113, 11324, 9310

Offset: 0

David A. Corneth, Aug 21 2024

			a(1) = 3 as 3 = 1 + 1 + 1 and 1 * 1 * 1 = 1 is a perfect cube.

David A. Corneth, Table of n, a(n) for n = 0..303
David A. Corneth, PARI program

Cf. A375580.

N:= 2*10^4:
V:= Array(1..N): count:= 0:
for x from 1 to N/3 do
  for y from x to (N-x)/2 do
     F:= ifactors(x*y)[2];
     b:= mul(t[1],t = select(s -> s[2] mod 3 = 2, F));
     c:= mul(t[1],t = select(s -> s[2] mod 3 = 1, F));
     for k from ceil((y/(b*c^2))^(1/3)) do
       s:= x+y+k^3 * b * c^2;
       if s > N then break fi;
       if s < x + 2*y then next fi;
       V[s]:= V[s]+1
od od od:
m:= max(V):
A:= Array(0..m): A[0]:= 1: count:= 1:
for i from 1 to N while count < m+1 do
  v:= V[i];
  if A[v] = 0 then A[v]:= i; count:= count+1 fi
od:
AL:= convert(V,list);
if not member(0,AL,'r') then r:= m+2 fi;
AL[1..r-1]; # Robert Israel,  Oct 21 2024, corrected Aug 22 2025

PARI
```
\\ See Corneth 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).

A382407 a(n) is the number of partitions n = x + y + z of positive integers such that xy + yz + x*z is a perfect square.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 3, 0, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 3, 0, 5, 1, 1, 2, 3, 3, 2, 1, 1, 3, 6, 1, 4, 2, 7, 4, 4, 0, 3, 5, 3, 4, 2, 1, 7, 2, 1, 5, 9, 3, 5, 3, 4, 1, 9, 2, 6, 3, 5, 6, 5, 4, 7, 5, 1, 5, 6, 3, 13, 7, 8, 4, 6, 0, 4, 4, 11, 5, 13, 2

Offset: 1

Felix Huber, Apr 04 2025

nonn

a(n) is the number of distinct cuboids with edge length 4*n whose surface area is half of a square.

Conjecture: a(k) = 0 iff k is an element of {2, 4, 8, 13} union A000244 union A005030.

			The a(14) = 3 partitions [x, y, z] are [1, 1, 12], [1, 4, 9] and [4, 4, 6] because 1*1 + 1*12 + 1*12 = 5^2, 1*4 + 4*9 + 1*9 = 7^2 and 4*4 + 4*6 + 4*6 = 8^2.

Felix Huber, Table of n, a(n) for n = 1..10000
Felix Huber, Maple program to calculate the partitions

Cf. A000244, A005030, A066955, A069905, A338939, A375512, A375576, A375580, A375731.

A382407:=proc(n)
    local a,x,y,z;
    a:=0;
    for x to n/3 do
        for y from x to (n-x)/2 do
            z:=n-x-y;
            if issqr(x*y+x*z+y*z) then
                a:=a+1
            fi
        od
    od;
    return a
end proc;
seq(A382407(n),n=1..87);

cp's OEIS Frontend

A375576 a(n) is the number of partitions n = x + y + z of positive integers such that x*y*z is a perfect square.

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A375785 a(n) is the number of distinct integer-sided cuboids having the same surface as a cube with edge length n.

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

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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.

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A375618 a(n) is the least positive integer k such that there are n partitions k = x + y + z of positive integers such that x * y * z is a perfect cube or -1 if no such positive integer exists.

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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.

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Formula

A382407 a(n) is the number of partitions n = x + y + z of positive integers such that x*y + y*z + x*z is a perfect square.

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Maple

A375576 a(n) is the number of partitions n = x + y + z of positive integers such that xyz is a perfect square.

A382407 a(n) is the number of partitions n = x + y + z of positive integers such that xy + yz + x*z is a perfect square.