A245440 Primes p == 1 (mod 4) such that p - floor(sqrt(p))^2 and 2p - floor(sqrt(2p))^2 are not squares.

353, 373, 449, 461, 521, 541, 593, 653, 673, 757, 769, 797, 821, 829, 941, 953, 1009, 1021, 1061, 1069, 1097, 1193, 1217, 1237, 1249, 1277, 1361, 1381, 1481, 1489, 1549, 1597, 1613, 1621, 1657, 1669, 1693, 1709, 1721, 1733, 1777, 1801, 1877, 1889, 1933, 1949

Offset: 1

Author

Thomas Ordowski, Jul 22 2014

Keywords

• nonn

Comments

Primes p of the form 4k+1 such that A053610(p) > 2 and A053610(2p) > 2.

Note that p = a^2 + b^2 and 2p = (a+b)^2 + (a-b)^2 is the only way. So according to the definition the greedy algorithm cannot give such the sums of two squares.

Interesting fact: a(n) = A145023(n) for all n < 25. Of course A145023 is a subsequence.

Primes p == 1 (mod 4) such that A245474(p) > 2.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10461 (first 46 terms from Thomas Ordowski and Colin Barker)

Crossrefs

Cf. A002144, A053610, A145016, A145023, A174806, A245474.

Programs

Magma

[p: p in PrimesUpTo(10000) | p mod 4 eq 1 and not IsSquare(p-Floor(Sqrt(p))^2) and not IsSquare(2*p-Floor(Sqrt(2*p))^2)]; // Vincenzo Librandi, Sep 19 2017

Mathematica

a245440Q[n_Integer] := If[
  And[PrimeQ[n] == True, Mod[n, 4] == 1],
  If[Or[IntegerQ[Sqrt[n - Floor[Sqrt[n]]^2]] == True,
    IntegerQ[Sqrt[2*n - Floor[Sqrt[2*n]]^2]] == True], False, True],
  False]; a245440[n_Integer] :=
Flatten[Position[Thread[a245440Q[Range[n]]],
   True]]; a245440[300000]; (* Michael De Vlieger, Aug 05 2014 *)

PARI

s=[]; forprime(p=2, 3000, if(p%4==1 && !issquare(p-floor(sqrt(p))^2) && !issquare(2*p-floor(sqrt(2*p))^2), s=concat(s, p))); s \\ Colin Barker, Jul 22 2014

Extensions

More terms from Colin Barker, Jul 22 2014