A369951 -id:A369951 - cp's OEIS frontend

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

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.
```

A370599 a(n) is the number of distinct triangles with integral side-lengths for which the perimeter 2*n divides the area.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 1, 1, 2, 2, 0, 0, 1, 1, 0, 4, 0, 1, 3, 0, 0, 1, 0, 0, 0, 0, 0, 3, 1, 2, 1, 0, 0, 3, 0, 0, 3, 2, 1, 4, 0, 2, 0, 3, 0, 2, 0, 0, 2, 2, 3, 1, 0, 2, 4, 1, 0, 8, 1, 0, 1

Offset: 1

Felix Huber, Mar 09 2024

If the perimeter 2*n of a triangle with integral edge-lengths divides its area A, then this also applies to a triangle stretched with positive integer k, because A*k^2/(2*n*k) = k*A/(2*n). Therefore a(d) <= a(n) for all positive divisors d of n and a(m) >= a(n) for all positive integer multiples m of n.

With an odd perimeter, according to Heron's formula the area A would have the form A = sqrt((2*k - 1)/8), where k is a positive integer. The area A would be irrational and the integer perimeter would not divide the area A. For this reason, only triangles with an even perimeter are considered in this sequence.

			a(18) = 1, because only the triangle (9, 10, 17) satisfies the condition: A/(2*n) = 36/36 = 1. (9, 10, 17) is one of the five triangles for which the perimeter is equal to the area (see A098030).
a(42) = 4, because exactly the 4 triangles (10, 35, 39) with A/(2*n) = 168/84 = 2, (14, 30, 40) with A/(2*n) = 168/84 = 2, (15, 34, 35) with A/(2*n) = 252/84 = 3 and (26, 28, 30) with A/(2*n) = 336/84 = 4 satisfy the condition.
a(426) = 0, because no triangle satisfies the condition. Therefore, a(n) = 0 for all n for which n*k = 426 for positive integers k.

Felix Huber, Table of n, a(n) for n = 1..500
Wikipedia, Heron's formula

Cf. A010814, A098030, A221837, A221838, A369951.

A370599 := proc(n) local u, v, w, A, q, i; i := 0; for u to floor(2/3*n) do for v from max(u, floor(n - u) + 1) to floor(n - 1/2*u) do w := 2*n - u - v; A := sqrt(n*(n - u)*(n - v)*(n - w)); if A = floor(A) then q := 1/2*A/n; if q = floor(q) then i := i + 1; end if; end if; end do; end do; return i; end proc;
seq(A370599(n), n = 1 .. 87);

a(n*k) >= a(n) for positive integers k.

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.
```

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A370599 a(n) is the number of distinct triangles with integral side-lengths for which the perimeter 2*n divides the area.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

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