A267763 Numbers whose base-3 representation is a square when read in base 10.

0, 1, 9, 16, 81, 100, 144, 235, 729, 784, 900, 961, 1296, 1369, 2115, 6561, 6724, 7056, 7225, 8100, 8649, 11664, 11881, 12321, 15985, 19035, 59049, 59536, 60516, 61009, 63504, 64009, 65025, 72900, 73441, 77841, 104976, 105625, 106929, 110889, 143865, 171315, 182428, 531441, 532900, 535824, 537289, 544644, 546121

Offset

1,3

Comments

Trivially includes powers of 9, since 9^k = 100..00_3 = 10^(2k) when read as a base-10 number. Moreover, for any a(n) in the sequence, 9*a(n) is also in the sequence. One could call "primitive" the terms not of this form; these would be 1, 16 = 121_3, 100 = 10201_3, 235 = 22201_3, 784 = 1002001_3, 961 = 1022121_3, ... These primitive terms include the subsequence 9^k + 2*3^k + 1, k > 0, which yields A033934 when written in base 3.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Mathematica

Select[Range[0, 600000], IntegerQ@Sqrt@FromDigits@IntegerDigits[#, 3] &] (* Vincenzo Librandi Dec 28 2016 *)

Prog

(PARI) is(n, b=3, c=10)=issquare(subst(Pol(digits(n, b)), x, c))
(Python)
A267763_list = [int(d, 3) for d in (str(i**2) for i in range(10**6)) if max(d) < '3'] # Chai Wah Wu, Mar 12 2016
(Magma) [n: n in [0..10^6] | IsSquare(Seqint(Intseq(n, 3)))]; // Vincenzo Librandi, Dec 28 2016

Crossrefs

Cf. A267764 - A267769 for bases 4 through 9. The base-2 analog is A000302 = powers of 4.

For a "prime" analog see also A235265, A235266, A152079, A235461 - A235482, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924.

Sequence in context: A203719 A198308 A167349 * A340164 A308248 A179307

Adjacent sequences: A267760 A267761 A267762 * A267764 A267765 A267766

Keyword

nonn,base

Author

M. F. Hasler, Jan 20 2016

Status

approved