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

1,1

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

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.

Formula

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

Example

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

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 *)

Prog

(Haskell)
a118886 n = a118886_list !! (n-1)
a118886_list = filter ((> 1) . a088534) a003136_list
-- Reinhard Zumkeller, Oct 30 2011
(PARI) is(n)=#bnfisintnorm(bnfinit(z^2+z+1), n) > 2;
select(is, vector(1500, j, j)) \\ Joerg Arndt, Jan 11 2015
(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

Crossrefs

Subsequence of Loeschian numbers A003136.

Complement of A198772 with respect to A003136.

Subsequences: A198773, A198774, A198775.

Cf. A024606, A118882.

Sequence in context: A157342 A230226 A178705 * A198773 A320633 A108164

Adjacent sequences: A118883 A118884 A118885 * A118887 A118888 A118889

Keyword

nonn

Author

Franklin T. Adams-Watters, May 03 2006

Status

approved