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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
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.
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
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 *)
CROSSREFS
Cf. (Analog for the square lattice) A230779, A001481, A004431, A002144, A004018, A084645.
Sequence in context: A024613 A024606 A074628 * A274437 A031194 A121058
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 28 04:13 EDT 2024. Contains 371235 sequences. (Running on oeis4.)