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A083854 Numbers that are squares, twice squares, three times squares, or six times squares, i.e., numbers whose squarefree part divides 6. 4
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 (list; graph; refs; listen; history; text; internal format)
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 triangles by its medians into small right-angled triangles; and 2*squares or 3*squares by recombining three or two of these small right-angled 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).

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.

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

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Last modified October 23 14:22 EDT 2019. Contains 328345 sequences. (Running on oeis4.)