A349202 -id:A349202 - cp's OEIS frontend

A348897 Numbers of the form (x + y)*(x^2 + y^2).

0, 1, 4, 8, 15, 27, 32, 40, 64, 65, 85, 108, 120, 125, 156, 175, 203, 216, 256, 259, 272, 320, 343, 369, 400, 405, 477, 500, 512, 520, 580, 585, 671, 680, 715, 729, 803, 820, 864, 888, 935, 960, 1000, 1080, 1105, 1111, 1157, 1248, 1261, 1331, 1372, 1400, 1417

Offset: 1

Peter Luschny, Nov 10 2021

nonn

Also numbers of the form (x - i*y)*(x + i*y)*(x + y).
Loeschian numbers of this form are A349200.
A349201 and A349202 are subsequences of this sequence.
Numbers of the form 1 + n + n^2 + n^3 (A053698) are a subsequence.
Numbers of the form n^3 + n^4 + n^5 + n^6 are a subsequence.
Numbers of the form 1 + n^2 + n^4 + n^6 (A059830) are a subsequence. - Bernard Schott, Nov 11 2021

			1010101 is in this sequence because 1010101 = (100 + 1)*(100^2 + 1^2).

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

Cf. A198063, A321500, A003136, A053698, A059830, A349200, A349201, A349202.

# Returns the terms less than or equal to b^3.
function A348897List(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
            push!(R, a)
    end end
unique!(sort!(R)) end
A348897List(12) |> println

# Returns the terms less than or equal to b^3.
A348897List := proc(b) local n, k, a, b3, R;
b3 := abs(b^3); R := {};
for n from 0 to b do for k from 0 to n do
    a := (n + k)*(n^2 + k^2);
    if a > b3 then break fi;
    R := R union {a};
od od; sort(R) end:
A348897List(12);

max = 2000;
xmax = max^(1/3) // Ceiling;
Table[(x + y) (x^2 + y^2), {x, 0, xmax}, {y, x, xmax}] // Flatten // Union // Select[#, # <= max&]& (* Jean-François Alcover, Oct 23 2023 *)

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

Original entry on oeis.org

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

Peter Luschny, Nov 10 2021

nonn

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.

			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).

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

Cf. A003136, A348897, A349202.

# 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

Intersection of A003136 and A348897.

cp's OEIS Frontend

A348897 Numbers of the form (x + y)*(x^2 + y^2).

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