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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 (list; graph; refs; listen; history; text; internal format)
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, A065720A036952, A065721 - A065727, A235394, A235395, A089971A020449, 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

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Last modified December 3 20:25 EST 2021. Contains 349468 sequences. (Running on oeis4.)