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

%I A245440 #57 Sep 08 2022 08:46:08
%S A245440 353,373,449,461,521,541,593,653,673,757,769,797,821,829,941,953,1009,
%T A245440 1021,1061,1069,1097,1193,1217,1237,1249,1277,1361,1381,1481,1489,
%U A245440 1549,1597,1613,1621,1657,1669,1693,1709,1721,1733,1777,1801,1877,1889,1933,1949
%N A245440 Primes p == 1 (mod 4) such that p - floor(sqrt(p))^2 and 2p - floor(sqrt(2p))^2 are not squares.
%C A245440 Primes p of the form 4k+1 such that A053610(p) > 2 and A053610(2p) > 2.
%C A245440 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.
%C A245440 Interesting fact: a(n) = A145023(n) for all n < 25. Of course A145023 is a subsequence.
%C A245440 Primes p == 1 (mod 4) such that A245474(p) > 2.
%H A245440 Michael De Vlieger, <a href="/A245440/b245440.txt">Table of n, a(n) for n = 1..10461</a> (first 46 terms from Thomas Ordowski and Colin Barker)
%t A245440 a245440Q[n_Integer] := If[
%t A245440   And[PrimeQ[n] == True, Mod[n, 4] == 1],
%t A245440   If[Or[IntegerQ[Sqrt[n - Floor[Sqrt[n]]^2]] == True,
%t A245440     IntegerQ[Sqrt[2*n - Floor[Sqrt[2*n]]^2]] == True], False, True],
%t A245440   False]; a245440[n_Integer] :=
%t A245440 Flatten[Position[Thread[a245440Q[Range[n]]],
%t A245440    True]]; a245440[300000]; (* _Michael De Vlieger_, Aug 05 2014 *)
%o A245440 (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
%o A245440 (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
%Y A245440 Cf. A002144, A053610, A145016, A145023, A174806, A245474.
%K A245440 nonn
%O A245440 1,1
%A A245440 _Thomas Ordowski_, Jul 22 2014
%E A245440 More terms from _Colin Barker_, Jul 22 2014