A118886 Numbers expressible as x^2 + x*y + y^2, 0 <= x <= y, in 2 or more ways.

49, 91, 133, 147, 169, 196, 217, 247, 259, 273, 301, 343, 361, 364, 399, 403, 427, 441, 469, 481, 507, 511, 532, 553, 559, 588, 589, 637, 651, 676, 679, 703, 721, 741, 763, 777, 784, 793, 817, 819, 868, 871, 889, 903, 931, 949, 961, 973, 988, 1027, 1029, 1036, 1057, 1083, 1092, 1099

Offset: 1

Author

Franklin T. Adams-Watters, May 03 2006

Keywords

• nonn

Comments

Squares of distances between two points in the triangular lattice in two or more nontrivially different ways.

Numbers whose prime factorization contains at least two (not necessarily distinct) primes congruent to 1 mod 6 and any prime factor congruent to 2 mod 3 has even multiplicity. Products of two elements of A024606.

If k is in the sequence then so is k * m^2 for m > 0. - David A. Corneth, Jun 21 2018

Examples

			a(2) = 91 = 1^2 + 1*9 + 9^2 = 5^2 + 5*6 + 6^2;
a(45) = 931 = 1^2+1*30+30^2 = 9^2+9*25+25^2 = 14^2+14*21+21^2;
a(97) = 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. - _Reinhard Zumkeller_, Oct 30 2011
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• A. Mazel, I. Stuhl, Y. Suhov, Hard-core configurations on a triangular lattice and Eisenstein primes, arXiv:1803.04041 [math.PR], 2018.

Crossrefs

Subsequence of Loeschian numbers A003136.

Complement of A198772 with respect to A003136.

Subsequences: A198773, A198774, A198775.

Cf. A024606, A118882.

Programs

Haskell

a118886 n = a118886_list !! (n-1)
a118886_list = filter ((> 1) . a088534) a003136_list
-- Reinhard Zumkeller, Oct 30 2011

Julia

function isA118886(n)
    n < 49 && return false
    n % 3 == 2 && return false
    M = Int(round(2*sqrt(n/3)))
    count = 0
    for y in 0:M, x in 0:y
        n == x^2 + y^2 + x*y && (count += 1)
        count == 2 && break
    end
    return count >= 2
end
A118886list(upto) = [n for n in 0:upto if isA118886(n)]
A118886list(1099) |> println # Peter Luschny, Mar 17 2018

Mathematica

amax = 2000; 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}];
A118886 = Select[Range[0, 3 xmax^2], # <= amax && f[#] > 1&] (* Jean-François Alcover, Jun 21 2018 *)

PARI

is(n)=#bnfisintnorm(bnfinit(z^2+z+1), n) > 2;
select(is, vector(1500,j,j)) \\ Joerg Arndt, Jan 11 2015

Formula

A088534(a(n)) > 1. - Reinhard Zumkeller, Oct 30 2011