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Numbers whose base-4 representation is a square when read in base 10.
3

%I #19 Jul 02 2019 02:11:34

%S 0,1,16,25,256,289,400,441,673,1761,1849,4096,4225,4624,4761,6400,

%T 6561,7056,7713,10768,13401,28176,29584,65536,66049,67600,68121,73984,

%U 74529,76176,76729,77985,102400,103041,104976,112896,113569,123408,150081,172288,214416,450816,473344,501433,519873

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

%C 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.

%H Chai Wah Wu, <a href="/A267764/b267764.txt">Table of n, a(n) for n = 1..10000</a>

%t Select[Range[1000], IntegerQ[Sqrt[FromDigits[IntegerDigits[#, 4]]]] &] (* _Alonso del Arte_, Jan 23 2016 *)

%o (PARI) is(n,b=4,c=10)=issquare(subst(Pol(digits(n,b)),x,c))

%o (Python)

%o 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

%Y Cf. A267763 - A267769 for bases 3 through 9. The base-2 analog is A000302 = powers of 4.

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

%K nonn,base

%O 1,3

%A _M. F. Hasler_, Jan 20 2016