A213158 -id:A213158 - cp's OEIS frontend

A225870 Nonnegative integers of the form xyz*(x+y-z) with integers x>=y>=z.

0, 1, 4, 9, 12, 16, 24, 25, 36, 40, 45, 49, 60, 64, 72, 81, 84, 100, 105, 112, 120, 121, 144, 160, 169, 180, 189, 192, 196, 216, 220, 225, 240, 252, 256, 264, 280, 289, 297, 300, 312, 324, 336, 352, 360, 361, 364, 384, 385, 396, 400, 420, 429, 432, 441, 480

Offset: 1

Michael Somos, May 18 2013

nonn

For n>=0 and n = x*y*z*(x+y-z) with integers x>=y>=z then we can even find nonnegative solutions (x,y,z). However, if we restrict to z>=0 then there are no solutions (x,y,z) in case n<0.
The negative integers of the form x*y*z*(x+y-z) with integers x>=y>=z are the negatives of A213158 and in that case z<0.
Nonnegative integers of the form (a^2-c^2)*(b^2-c^2) with integers a>=b>=c.
Note that we must allow c<0 to represent n=12, 24, 40, ....
The negative integers of the form (a^2-c^2)*(b^2-c^2) with integers a>=b>=c are the negatives of A213158.

			12 = (1)*(-2)*(-3)*((1)+(-2)-(-3)) with (x,y,z) = (1,-2,-3).
12 = 2*2*1*(2+2-1) with (x,y,z) = (2,2,1).
12 = ((0)^2-(-2)^2)*((-1)^2-(-2)^2) with (a,b,c) = (0,-1,-2).
12 = ((1)^2-(-2)^2)*((0)^2-(-2)^2) with (a,b,c) = (1,0,-2).

Cf. A213158.

{isa(n) = forvec( v = vector(3, i, [0, ceil(n^(1/2))]), if( n == v[1] * v[2] * v[3] * (v[3] + v[2] - v[1]), return(1)), 1)}

A376166 Integers m whose decimal representation can be split into integers a,b,c such that the triangle with sides a,b,c has area sqrt(m).

Original entry on oeis.org

459756, 80312850756

Offset: 1

Max Alekseyev, Sep 13 2024

Instances of b and c in m can have leading zeros.
Each term is congruent to 0, 8, 16, 28, 36, 44, 48, 56, 68, 76, 88, or 96 modulo 100.
The length of c in m for any further term should be at least 6.

			45|97|56 is the squared area of a triangle with sides 45, 97, 56.
803|1285|0756 is the squared area of a triangle with sides 803, 1285, 756.

Diego Rattaggi, Just a triangle, X, 2019.

Subsequence of A213158.

a(1) was found by Diego Rattaggi.

cp's OEIS Frontend

A225870 Nonnegative integers of the form xyz*(x+y-z) with integers x>=y>=z.

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A376166 Integers m whose decimal representation can be split into integers a,b,c such that the triangle with sides a,b,c has area sqrt(m).

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cp's OEIS Frontend

A225870 Nonnegative integers of the form x*y*z*(x+y-z) with integers x>=y>=z.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A376166 Integers m whose decimal representation can be split into integers a,b,c such that the triangle with sides a,b,c has area sqrt(m).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A225870 Nonnegative integers of the form xyz*(x+y-z) with integers x>=y>=z.