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

%I A349200 #14 Dec 25 2021 02:37:37
%S A349200 0,1,4,27,64,108,156,175,256,259,343,400,729,1261,1372,1417,1728,1875,
%T A349200 2197,2916,3439,3492,3667,4096,4212,4579,4725,6175,6859,6912,6993,
%U A349200 7104,7825,8112,8125,8425,8788,9261,9264,9325,9925,9984,10800,11200,11425,11712
%N A349200 Loeschian numbers of the form (x + y)*(x^2 + y^2).
%C A349200 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.
%H A349200 Peter Luschny, <a href="/A349200/b349200.txt">Table of n, a(n) for n = 1..1200</a>
%F A349200 Intersection of A003136 and A348897.
%e A349200 729  = 27^2 + 27*0 + 0^2   = (9 + 0)*(9^2 + 0^2).
%e A349200 3492 = 48^2 + 48*18 + 18^2 = (13 + 5)*(13^2 + 5^2).
%e A349200 3667 = 53^2 + 53*13 + 13^2 = (12 + 7)*(12^2 + 7^2).
%o A349200 (Julia) # Returns the terms less than or equal to b^3.
%o A349200 # Uses the function isA003136 from A003136.
%o A349200 function A349200List(b)
%o A349200     b3 = b^3; R = [0]
%o A349200     for n in 1:b
%o A349200         for k in 0:n
%o A349200             a = (n + k) * (n^2 + k^2)
%o A349200             a > b3 && break
%o A349200             isA003136(a) && push!(R, a)
%o A349200     end end
%o A349200 sort(R) end
%o A349200 A349200List(24) |> println
%Y A349200 Cf. A003136, A348897, A349202.
%K A349200 nonn
%O A349200 1,3
%A A349200 _Peter Luschny_, Nov 10 2021