A176189 Natural numbers whose squares have only 0's and 1's in base 3.

1, 2, 3, 6, 9, 11, 16, 18, 19, 27, 29, 33, 48, 54, 55, 57, 81, 83, 87, 99, 143, 144, 162, 163, 165, 171, 243, 245, 249, 261, 262, 297, 421, 429, 432, 451, 486, 487, 489, 495, 513, 729, 731, 735, 747, 783, 786, 889, 891, 1263, 1287, 1296, 1331, 1342, 1353, 1458

Offset: 1

Author

Maciej Ireneusz Wilczynski, Apr 11 2010

Keywords

• base
• easy
• nonn

Comments

If 3 divides a(n) then a(n)/3 also appears in this sequence. Also the inverse is true: if a(n) appears, then (3^k)*a(n), for all k>=0, appears as well.

Note that a(n) usually does not consist only of 0's and 1's - it can be shown that in this case a(n)=3^k, for some k>=0.

So, a(n)^2 belongs to A005836. - Michel Marcus, Nov 12 2012

Examples

			For n=16 we have 16^2=256="100111" (in base 3). Also (16*3)^2="10011100", (16*3^2)^2="1001110000", etc.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• K. Mahler, The representation of squares to the base 3, Acta Arith. Vol. 53, Issue 1 (1989), p. 99-106.

Crossrefs

Cf. A005836.

Programs

Mathematica

Select[Range[1200], Max[IntegerDigits[ #^2, 3]] == 1 &]

Python

from gmpy2 import digits
def ok(n): return "2" not in digits(n*n, 3)
print([k for k in range(1, 1500) if ok(k)]) # Michael S. Branicky, Jun 07 2023