A145022 Primes p of the form 4k+1 for which s=2 is the least positive integer such that sp-(floor(sqrt(sp)))^2 is a square.

41, 61, 89, 113, 149, 157, 181, 193, 233, 269, 277, 313, 317, 337, 389, 421, 433, 557, 569, 613, 617, 709, 761, 773, 853, 881, 929, 937, 1013, 1109, 1117, 1129, 1201, 1213, 1301, 1409, 1429, 1553, 1637, 1741, 1753, 1861, 1873, 1901, 1997, 2113, 2137, 2153

Offset: 1

Author

Vladimir Shevelev, Sep 29 2008

Keywords

• nonn

Examples

			a(1)=41 since p=41 is the least prime of the form 4k+1 for which p-(floor(sqrt(p)))^2 is not a square, but 2p-(floor(sqrt(2p)))^2 is a square (for p=41 it is 1).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A145016.

Programs

Maple

filter:= proc(p)
  if not isprime(p) then return false fi;
  if issqr(p-floor(sqrt(p))^2) then return false fi;
  issqr(2*p-floor(sqrt(2*p))^2)
end proc:
select(filter, [seq(p,p=1..10000,4)]); # Robert Israel, Dec 04 2018

Mathematica

sQ[n_] := IntegerQ[Sqrt[n - (Floor[Sqrt[n]])^2]]; aQ[n_] := Mod[n, 4] == 1 && PrimeQ[n] && !sQ[n] && sQ[2n]; Select[Range[2200], aQ] (* Amiram Eldar, Dec 04 2018 *)