A357019 - OEIS

A357019

a(n) is the largest possible x in n = x^2 - x*y + y^2 with integers x > y >= 0, or 0 if n cannot be expressed in this form.

%I #16 Sep 12 2022 17:14:19

%S 0,1,0,2,2,0,0,3,0,3,0,0,4,4,0,0,4,0,0,5,0,5,0,0,0,5,0,6,6,0,0,6,0,0,

%T 0,0,6,7,0,7,0,0,0,7,0,0,0,0,8,8,0,0,8,0,0,0,0,8,0,0,0,9,0,9,8,0,0,9,

%U 0,0,0,0,0,9,0,10,10,0,0,10,0,9,0,0,10,0

%N a(n) is the largest possible x in n = x^2 - x*y + y^2 with integers x > y >= 0, or 0 if n cannot be expressed in this form.

%o (PARI) dloesch(n) = {my(L=List()); foreach([-1,1], qs, my (D=qfbsolve(Qfb(1,qs,1), factor(n), 3), dnp=#D); for (k=1, dnp, if(D[k][1]^2+D[k][2]^2-abs(D[k][1]*D[k][2])==n, listput (L, [abs(D[k][1]),abs(D[k][2])])))); Set(L)};

%o for (k=1, 85, my(D=dloesch(k), d=#D, m=0); for (j=1, d, m=max(m,D[j][1]));print1(m,", "))

%o (Python)

%o from sympy.abc import x, y

%o from sympy.solvers.diophantine.diophantine import diop_quadratic

%o def A357019(n): return max((a for a,b in diop_quadratic(x*(x-y)+y**2-n)),default=0) # _Chai Wah Wu_, Sep 12 2022

%Y Cf. A002324, A003136, A133388.

%K nonn

%O 0,4

%A _Hugo Pfoertner_, Sep 10 2022