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

0,3

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

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

Mathematica

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

Crossrefs

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

Sequence in context: A296991 A097755 A301704 * A275199 A003586 A114334

Adjacent sequences: A083851 A083852 A083853 * A083855 A083856 A083857

Keyword

nonn

Author

Henry Bottomley, May 06 2003

Status

approved