A145017 Squarefree positive integers k for which k-(floor(sqrt(k)))^2 is a perfect square.

1, 2, 5, 10, 13, 17, 26, 29, 34, 37, 53, 58, 65, 73, 82, 85, 97, 101, 109, 122, 130, 137, 145, 170, 173, 178, 185, 194, 197, 205, 221, 226, 229, 241, 257, 265, 281, 290, 293, 298, 305, 314, 349, 362, 365, 370, 377, 386, 397, 401, 409, 442, 445, 457, 466, 485, 493

Offset

1,2

Comments

If an odd prime p divides a(n) then it has the form 4k+1.

Conjecture. For every n>=1 there exist infinitely many primes p of the form 4k+1 for which for a(n) > 1 we have s*p-(floor(sqrt(s*p)))^2 is not a perfect square for s=1,...,a(n)-1 while a(n)*p-(floor(sqrt(a(n)p))^2 is a perfect square. (See A145016(s=1) and A145022, A145023, A145047, A145048, A145149, A145050 correspondingly for s=2, s=5, s=10, s=13, s=17, s=26.) - Vladimir Shevelev, Sep 30 2008

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Mathematica

Select[Range@ 500, And[SquareFreeQ@ #, IntegerQ@ Sqrt[# - Floor[Sqrt@ #]^2]] &] (* Michael De Vlieger, Jan 12 2020 *)

Prog

(PARI) is(n)={issquarefree(n) && issquare(n-sqrtint(n)^2)} \\ Andrew Howroyd, Jan 12 2020

Crossrefs

Cf. A005117, A020893, A145016.

Cf. A145022, A045023, A145047, A145048, A145049, A145050.

Sequence in context: A226828 A020893 A281292 * A031396 A003814 A003654

Adjacent sequences: A145014 A145015 A145016 * A145018 A145019 A145020

Keyword

nonn

Author

Vladimir Shevelev, Sep 29 2008

Extensions

Missing a(40) inserted and terms a(42) and beyond from Andrew Howroyd, Jan 12 2020

Status

approved