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 A118886 Numbers expressible as x^2 + x*y + y^2, 0 <= x <= y, in 2 or more ways. 11
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified October 20 22:05 EDT 2019. Contains 328291 sequences. (Running on oeis4.)