A198775 Numbers having exactly four representations by the quadratic form x^2+xy+y^2 with 0<=x<=y.

1729, 2821, 3367, 3913, 4123, 4459, 4921, 5187, 5551, 5719, 6097, 6517, 6643, 6916, 7189, 7657, 8029, 8113, 8463, 8827, 8911, 9139, 9331, 9373, 9709, 9919, 10101, 10507, 10621, 10633, 11137, 11284, 11557, 11739, 12369, 12649, 12691, 12901, 13237, 13377

Offset: 1

Author

Reinhard Zumkeller, Oct 30 2011

Keywords

• nonn

Comments

A088534(a(n)) = 4; subsequence of A118886, see also A003136.

Examples

			a(1) = 1729 = 3^2+3*40+40^2 = 8^2+8*37+37^2 = 15^2+15*32+32^2 = 23^2+23*25+25^2, A088534(1729) = 4;
a(10) = 5719 = 5^2+5*73+73^2 = 15^2+15*67+67^2 = 18^2+18*65+65^2 = 37^2+37*50+50^2, A088534(5719) = 4;
a(100) = 23779 = 17^2+17*145+145^2 = 30^2+30*137+137^2 = 50^2+50*123+123^2 = 85^2+85*93+93^2, A088534(23779) = 4.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..250 from Reinhard Zumkeller).

Crossrefs

Cf. A198772, A198773, A198774.

Programs

Haskell

a198775 n = a198775_list !! (n-1)
a198775_list = filter ((== 4) . a088534) a003136_list

Mathematica

amax = 20000; xmax = Sqrt[amax] // Ceiling; Clear[f]; f[_] = 0; Do[q = x^2 + x y + y^2; f[q] = f[q] + 1, {x, 0, xmax}, {y, x, xmax}];
A198775 = Select[Range[0, 3 xmax^2], # <= amax && f[#] == 4&] (* Jean-François Alcover, Jun 21 2018 *)

Python

from itertools import count, islice
def A198775_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        c = 0
        for y in range(n+1):
            if c > 4 or y**2 > n:
                break
            for x in range(y+1):
                z = x*(x+y)+y**2
                if z > n:
                    break
                elif z == n:
                    c += 1
                    if c > 4:
                        break
        if c == 4:
            yield n
A198775_list = list(islice(A198775_gen(),10)) # Chai Wah Wu, May 16 2022