A281898 Numbers k such that k - floor(sqrt(k))^2 and 2k - floor(sqrt(2k))^2 are both squares.

0, 1, 2, 4, 5, 8, 10, 13, 17, 20, 25, 29, 34, 40, 45, 50, 58, 65, 80, 85, 97, 100, 125, 130, 145, 170, 185, 200, 221, 225, 250, 260, 265, 290, 325, 340, 365, 377, 400, 425, 445, 450, 485, 493, 520, 530, 545, 580, 625, 650, 680, 685, 730, 754, 765, 785, 800, 841, 845, 890, 900

Offset: 1

Author

Thomas Ordowski, Feb 01 2017

Keywords

• nonn

Comments

The sequence is infinite: it contains an infinite subsequence { A000129(2*k)^2 + 1, k>=0 }. - Max Alekseyev, Feb 01 2017

Also A000129(2k+1)^2 is a subsequence.

There are precisely six primes in this sequence: 2, 5, 13, 17, 29, and 97.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000129, A076218 is a subsequence, A145016.

Programs

Mathematica

Select[Range[0, 900], Times @@ Boole@ Map[IntegerQ@ Sqrt@ # &, # - Floor[Sqrt@ #]^2 &@ {#, 2 #}] == 1 &] (* Michael De Vlieger, Feb 02 2017 *)
Select[Range[0,1000],AllTrue[{Sqrt[#-Floor[Sqrt[#]]^2],Sqrt[2#-Floor[ Sqrt[ 2#]]^2]},IntegerQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 25 2020 *)

PARI

is(n)=issquare(n-sqrtint(n)^2) && issquare(2*n-sqrtint(2*n)^2) \\ Charles R Greathouse IV, Feb 01 2017

Extensions

More terms from Altug Alkan, Feb 01 2017