A230781 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.

1, 3, 4, 9, 12, 16, 25, 27, 36, 48, 64, 75, 81, 100, 108, 121, 144, 192, 225, 243, 256, 289, 300, 324, 363, 400, 432, 484, 529, 576, 625, 675, 729, 768, 841, 867, 900, 972, 1024, 1089, 1156, 1200, 1296, 1452, 1587, 1600, 1681, 1728, 1875, 1936, 2025, 2116, 2187, 2209, 2304, 2500

Offset: 1

Author

Jean-Christophe Hervé, Nov 23 2013

Keywords

• nonn

Comments

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

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.

Examples

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

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 (first 413 terms from Jean-Christophe Hervé)
• August Lösch, Economics of Location (1954), see pp. 117f.
• U. P. Nair, Elementary results on the binary quadratic form a^2+ab+b^2
• Kenneth Shum's scrapbook, Distance distribution in the hexagonal packing
• Index entries for sequences related to A2 = hexagonal = triangular lattice

Crossrefs

Cf. A003136, A003627, A024606.

Programs

Mathematica

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 *)