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A118886
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Numbers expressible as x^2 + x*y + y^2, 0 <= x <= y, in 2 or more ways.
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11
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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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
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MATHEMATICA
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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}];
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PROG
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(Haskell)
a118886 n = a118886_list !! (n-1)
a118886_list = filter ((> 1) . a088534) a003136_list
(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)]
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CROSSREFS
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Subsequence of Loeschian numbers A003136.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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