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

1,2

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

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.

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A005836.

Sequence in context: A227000 A338546 A358130 * A057127 A224449 A007780

Adjacent sequences: A176186 A176187 A176188 * A176190 A176191 A176192

Keyword

base,easy,nonn

Author

Maciej Ireneusz Wilczynski, Apr 11 2010

Status

approved