A349200 Loeschian numbers of the form (x + y)*(x^2 + y^2).

0, 1, 4, 27, 64, 108, 156, 175, 256, 259, 343, 400, 729, 1261, 1372, 1417, 1728, 1875, 2197, 2916, 3439, 3492, 3667, 4096, 4212, 4579, 4725, 6175, 6859, 6912, 6993, 7104, 7825, 8112, 8125, 8425, 8788, 9261, 9264, 9325, 9925, 9984, 10800, 11200, 11425, 11712

Offset: 1

Author

Peter Luschny, Nov 10 2021

Keywords

• nonn

Comments

k is in this sequence if there exist numbers x, y, v, w such that k = x^2 + x*y + y^2 = (v + w)*(v^2 + w^2). We call (x, y, v, w) a witness of k. k can have different witnesses.

Examples

			729  = 27^2 + 27*0 + 0^2   = (9 + 0)*(9^2 + 0^2).
3492 = 48^2 + 48*18 + 18^2 = (13 + 5)*(13^2 + 5^2).
3667 = 53^2 + 53*13 + 13^2 = (12 + 7)*(12^2 + 7^2).
		

Links

• Peter Luschny, Table of n, a(n) for n = 1..1200

Crossrefs

Cf. A003136, A348897, A349202.

Programs

Julia

# Returns the terms less than or equal to b^3.
# Uses the function isA003136 from A003136.
function A349200List(b)
    b3 = b^3; R = [0]
    for n in 1:b
        for k in 0:n
            a = (n + k) * (n^2 + k^2)
            a > b3 && break
            isA003136(a) && push!(R, a)
    end end
sort(R) end
A349200List(24) |> println

Formula

Intersection of A003136 and A348897.