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.

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, 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, 0, 0, 0, 0, 0, 9, 0, 10, 10, 0, 0, 10, 0, 9, 0, 0, 10, 0

Offset: 0

Author

Hugo Pfoertner, Sep 10 2022

Keywords

• nonn

Crossrefs

Cf. A002324, A003136, A133388.

Programs

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

Python

from sympy.abc import x, y
from sympy.solvers.diophantine.diophantine import diop_quadratic
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