A083854 Numbers that are squares, twice squares, three times squares, or six times squares, i.e., numbers whose squarefree part divides 6.

0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 25, 27, 32, 36, 48, 49, 50, 54, 64, 72, 75, 81, 96, 98, 100, 108, 121, 128, 144, 147, 150, 162, 169, 192, 196, 200, 216, 225, 242, 243, 256, 288, 289, 294, 300, 324, 338, 361, 363, 384, 392, 400, 432, 441, 450, 484, 486, 507

Offset: 0

Author

Henry Bottomley, May 06 2003

Keywords

• nonn

Comments

It is simple to divide equilateral triangles into these numbers of congruent parts: squares by making smaller equilateral triangles; 6*squares by dividing each small equilateral triangle by its medians into small right triangles; and 2*squares or 3*squares by recombining three or two of these small right triangles.

Links

• Ivan Neretin, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000290, A007913, A001105, A028982, A033428, A033581, A083855, A195055.

Programs

Mathematica

mx = 23; Sort@Select[Flatten@Table[{1, 2, 3, 6} n^2, {n, mx}], # <= mx^2 &] (* Ivan Neretin, Nov 08 2016 *)

Formula

a(n) is bounded below by 0.137918...*n^2 where 0.137918... = 3*(3-2*sqrt(2))*(2-sqrt(3)); the error appears to be O(n).

Sum_{n>=1} 1/a(n) = Pi^2/3 (A195055). - Amiram Eldar, Dec 19 2020