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A267763
Numbers whose base-3 representation is a square when read in base 10.
7
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.
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.
Sequence in context: A198308 A375501 A167349 * A340164 A308248 A179307
KEYWORD
nonn,base
AUTHOR
M. F. Hasler, Jan 20 2016
STATUS
approved