login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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
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.
Sequence in context: A157342 A230226 A178705 * A198773 A320633 A108164
KEYWORD
nonn
AUTHOR
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)