A230781 - OEIS

%I #14 Feb 19 2020 14:29:15

%S 1,3,4,9,12,16,25,27,36,48,64,75,81,100,108,121,144,192,225,243,256,

%T 289,300,324,363,400,432,484,529,576,625,675,729,768,841,867,900,972,

%U 1024,1089,1156,1200,1296,1452,1587,1600,1681,1728,1875,1936,2025,2116,2187,2209,2304,2500

%N Squared radii of circles centered at a grid point in a triangular lattice hitting exactly 6 points. Squares or triple of squares that are not expressible as x^2+xy+y^2 with y > x > 0.

%C Numbers without prime factor of form 6k+1 and without prime factor of form 3k+2 to an odd multiplicity.

%C The sequence is closed under multiplication. Primitive elements are 3 and square of primes of form 3k+2 (A003627). Sequence A003136 (Loeschian numbers) is the union of {0}, this sequence and A024606 (numbers of form x^2+xy+y^2 with y > x > 0). These 4 sequences are all closed under multiplication.

%H Ray Chandler, <a href="/A230781/b230781.txt">Table of n, a(n) for n = 1..10000</a> (first 413 terms from Jean-Christophe Hervé)

%H August Lösch, <a href="http://archive.org/stream/economicsoflocat00ls#page/116/mode/2up">Economics of Location</a> (1954), see pp. 117f.

%H U. P. Nair, <a href="http://arXiv.org/abs/math.NT/0408107">Elementary results on the binary quadratic form a^2+ab+b^2</a>

%H Kenneth Shum's scrapbook, <a href="http://home.ie.cuhk.edu.hk/~wkshum/wordpress/?p=1003">Distance distribution in the hexagonal packing</a>

%H <a href="/index/Aa#A2">Index entries for sequences related to A2 = hexagonal = triangular lattice</a>

%e 49 is not in the sequence because 49 = 3^2+3*5+5^2.

%t selQ[1] = True; selQ[n_] := Module[{f = FactorInteger[n]}, FreeQ[f, {p_, q_} /; Mod[p, 6] == 1 || Mod[p, 3] == 2 && OddQ[q]]]; Select[Range[2500], selQ] (* _Jean-François Alcover_, Nov 25 2013, after first comment *)

%Y Cf. A003136, A003627, A024606.

%K nonn

%O 1,2

%A _Jean-Christophe Hervé_, Nov 23 2013