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.

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

Author

Felix Huber, Apr 04 2025

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

Maple

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);