A267764 Numbers whose base-4 representation is a square when read in base 10.

0, 1, 16, 25, 256, 289, 400, 441, 673, 1761, 1849, 4096, 4225, 4624, 4761, 6400, 6561, 7056, 7713, 10768, 13401, 28176, 29584, 65536, 66049, 67600, 68121, 73984, 74529, 76176, 76729, 77985, 102400, 103041, 104976, 112896, 113569, 123408, 150081, 172288, 214416, 450816, 473344, 501433, 519873

Offset

1,3

Comments

Trivially includes powers of 16, since 16^k = 100..00_4 = 10^(2k) when read as a base-10 number. Moreover, for any a(n) in the sequence, 16*a(n) is also in the sequence. One could call "primitive" the terms not of this form, these would be 1, 25 = 121_4, 289 = 10201_4, 441 = 12321_4, 673 = 22201_4, 1761 = 123201_4, ... These primitive terms include the subsequence 16^k + 2*4^k + 1 = (4^k+1)^2, k > 0, which yields A033934 when written in base 4.

Links

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

Mathematica

Select[Range[1000], IntegerQ[Sqrt[FromDigits[IntegerDigits[#, 4]]]] &] (* Alonso del Arte, Jan 23 2016 *)

Prog

(PARI) is(n, b=4, c=10)=issquare(subst(Pol(digits(n, b)), x, c))
(Python)
A267764_list = [int(d, 4) for d in (str(i**2) for i in range(10**6)) if max(d) < '4'] # Chai Wah Wu, Feb 23 2016

Crossrefs

Cf. A267763 - A267769 for bases 3 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: A061101 A166672 A167358 * A264588 A291093 A291966

Adjacent sequences: A267761 A267762 A267763 * A267765 A267766 A267767

Keyword

nonn,base

Author

M. F. Hasler, Jan 20 2016

Status

approved