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 A232436 Numbers which are uniquely decomposable into x^2+xy+y^2, the unique decomposition being with two distinct nonzero x and y. 3
 7, 13, 19, 21, 28, 31, 37, 39, 43, 52, 57, 61, 63, 67, 73, 76, 79, 84, 93, 97, 103, 109, 111, 112, 117, 124, 127, 129, 139, 148, 151, 156, 157, 163, 171, 172, 175, 181, 183, 189, 193, 199, 201, 208, 211, 219, 223, 228, 229, 237, 241, 244, 252, 268 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS These are the primitive elements of A024606, the integers which are expressible as x^2 + xy + y^2 with distinct nonzero x and y. 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. 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. From a(n) = 3^a*p*q^2, it is easily seen that sigma(a(n)) = 2 mod 6, 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. 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. LINKS Jean-Christophe Hervé, Table of n, a(n) for n = 1..6364 A. Mazel, I. Stuhl, Y. Suhov, Hard-core configurations on a triangular lattice and Eisenstein primes, arXiv:1803.04041 [math.PR], 2018. G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Terms are obtained by the products A230781(k)*A002476(p) for k, p > 0, ordered by increasing values. A004016(a(n))=12. EXAMPLE 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). MATHEMATICA selQ = False; selQ[n_] := Module[{f = FactorInteger[n]}, Count[f, {p_ /; Mod[p, 6] == 1, 1}] == 1 && FreeQ[f, {p_ /; Mod[p, 3] == 2, _?OddQ}]]; Select[Range, selQ] (* Jean-François Alcover, Nov 25 2013, after 3rd comment *) CROSSREFS Cf. A003136, A024606, A002476, A232437. Cf. A004016, A034020 (A004016 = 0), A230781 (A004016 = 6). Cf. (Analog for the square lattice) A230779, A001481, A004431, A002144, A004018, A084645. Sequence in context: A024613 A024606 A074628 * A274437 A031194 A121058 Adjacent sequences:  A232433 A232434 A232435 * A232437 A232438 A232439 KEYWORD nonn AUTHOR Jean-Christophe Hervé, Nov 23 2013 STATUS approved

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Last modified December 7 03:00 EST 2019. Contains 329836 sequences. (Running on oeis4.)