cp's OEIS Frontend

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

%I A083854 #20 Dec 19 2020 03:36:12
%S A083854 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,
%T A083854 98,100,108,121,128,144,147,150,162,169,192,196,200,216,225,242,243,
%U A083854 256,288,289,294,300,324,338,361,363,384,392,400,432,441,450,484,486,507
%N A083854 Numbers that are squares, twice squares, three times squares, or six times squares, i.e., numbers whose squarefree part divides 6.
%C A083854 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.
%H A083854 Ivan Neretin, <a href="/A083854/b083854.txt">Table of n, a(n) for n = 0..10000</a>
%F A083854 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).
%F A083854 Sum_{n>=1} 1/a(n) = Pi^2/3 (A195055). - _Amiram Eldar_, Dec 19 2020
%t A083854 mx = 23; Sort@Select[Flatten@Table[{1, 2, 3, 6} n^2, {n, mx}], # <= mx^2 &] (* _Ivan Neretin_, Nov 08 2016 *)
%Y A083854 Cf. A000290, A007913, A001105, A028982, A033428, A033581, A083855, A195055.
%K A083854 nonn
%O A083854 0,3
%A A083854 _Henry Bottomley_, May 06 2003