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%I #27 Feb 20 2020 06:57:57
%S 7,13,19,21,28,31,37,39,43,52,57,61,63,67,73,76,79,84,93,97,103,109,
%T 111,112,117,124,127,129,139,148,151,156,157,163,171,172,175,181,183,
%U 189,193,199,201,208,211,219,223,228,229,237,241,244,252,268
%N Numbers which are uniquely decomposable into x^2+xy+y^2, the unique decomposition being with two distinct nonzero x and y.
%C These are the primitive elements of A024606, the integers which are expressible as x^2 + xy + y^2 with distinct nonzero x and y.
%C As a subsequence of A003136 (Loeschian numbers), the sequence is related with the triangular lattice: circles with radius sqrt(a(n)) centered at a grid point in this lattice hit exactly 12 points, cf. A004016.
%C Numbers with exactly one prime factor of form 6k+1 with multiplicity one and no prime factor of form 3k+2 with odd multiplicity, that is a(n) is of form 3^a*p*q^2, with a>=0, p a prime of form 6k+1, and q an integer with all its prime factors of form 3k+2. There is thus no square in the sequence.
%C From a(n) = 3^a*p*q^2, it is easily seen that sigma(a(n)) = 2 mod 6,
%C thus this sequence is a subsequence of A074628: the two sequences are equal up to a(308) = 1723; then A074628(309)= 1729 = a(1)*a(2)*a(3), the famous Ramanujan's taxi number, and a(309) = A074628(310) = 1731.
%C The square of these numbers is also uniquely decomposable into the form x^2 + xy + y^2 with x and y > 0, thus this sequence is a subsequence of A232437.
%H Ray Chandler, <a href="/A232436/b232436.txt">Table of n, a(n) for n = 1..10000</a> (first 6364 terms from Jean-Christophe Hervé)
%H A. Mazel, I. Stuhl, Y. Suhov, <a href="https://arxiv.org/abs/1803.04041">Hard-core configurations on a triangular lattice and Eisenstein primes</a>, arXiv:1803.04041 [math.PR], 2018.
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H <a href="/index/Aa#A2">Index entries for sequences related to A2 = hexagonal = triangular lattice</a>
%F Terms are obtained by the products A230781(k)*A002476(p) for k, p > 0, ordered by increasing values.
%F A004016(a(n))=12.
%e a(1)= 7 = 2^2+2+1, a(2)= 13 = 3^2+3+1. However 3 = 1+1+1 and 4 = 2^2+0*2+0 are not in the sequence because the unique decomposition of these numbers is not with two distinct nonzero numbers; 49, 147 are also excluded because there are two decompositions of these numbers (including one with equal or zero components x and y).
%t r[k_] := Reduce[x != 0 && y != 0 && x != y && k == x^2 + x y + y^2, {x, y}, Integers];
%t selQ[k_] := If[IntegerQ[Sqrt[k]], False, Which[rk = r[k]; rk === False, False, rk[[0]] === And && Length[rk] == 2, True, rk[[0]] === Or && Length[rk] == 12, True, True, False]];
%t Select[Range[1000], selQ] (* _Jean-François Alcover_, Feb 20 2020 *)
%Y Cf. A003136, A024606, A002476, A232437.
%Y Cf. A004016, A034020 (A004016 = 0), A230781 (A004016 = 6).
%Y Cf. (Analog for the square lattice) A230779, A001481, A004431, A002144, A004018, A084645.
%K nonn
%O 1,1
%A _Jean-Christophe Hervé_, Nov 23 2013
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