A176189 - OEIS

%I #16 Jun 08 2023 02:04:25

%S 1,2,3,6,9,11,16,18,19,27,29,33,48,54,55,57,81,83,87,99,143,144,162,

%T 163,165,171,243,245,249,261,262,297,421,429,432,451,486,487,489,495,

%U 513,729,731,735,747,783,786,889,891,1263,1287,1296,1331,1342,1353,1458

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

%C 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.

%C 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.

%C So, a(n)^2 belongs to A005836. - _Michel Marcus_, Nov 12 2012

%H Alois P. Heinz, <a href="/A176189/b176189.txt">Table of n, a(n) for n = 1..1000</a>

%H K. Mahler, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa53/aa5316.pdf">The representation of squares to the base 3</a>, Acta Arith. Vol. 53, Issue 1 (1989), p. 99-106.

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

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

%o (Python)

%o from gmpy2 import digits

%o def ok(n): return "2" not in digits(n*n, 3)

%o print([k for k in range(1, 1500) if ok(k)]) # _Michael S. Branicky_, Jun 07 2023

%Y Cf. A005836.

%K base,easy,nonn

%O 1,2

%A _Maciej Ireneusz Wilczynski_, Apr 11 2010