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A232436
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Numbers which are uniquely decomposable into x^2+xy+y^2, the unique decomposition being with two distinct nonzero x and y.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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Ray Chandler, Table of n, a(n) for n = 1..10000 (first 6364 terms from Jean-Christophe Hervé)
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
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to A2 = hexagonal = triangular lattice
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FORMULA
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Terms are obtained by the products A230781(k)*A002476(p) for k, p > 0, ordered by increasing values.
A004016(a(n))=12.
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EXAMPLE
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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).
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MATHEMATICA
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r[k_] := Reduce[x != 0 && y != 0 && x != y && k == x^2 + x y + y^2, {x, y}, Integers];
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]];
Select[Range[1000], selQ] (* Jean-François Alcover, Feb 20 2020 *)
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Jean-Christophe Hervé, Nov 23 2013
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STATUS
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approved
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