A245440 - OEIS

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

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OFFSET

1,1

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)

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 *)

PROG

(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

(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

CROSSREFS

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

Sequence in context: A279583 A156177 A104160 * A145023 A343714 A343715

Adjacent sequences: A245437 A245438 A245439 * A245441 A245442 A245443

KEYWORD

nonn

AUTHOR

Thomas Ordowski, Jul 22 2014

EXTENSIONS

More terms from Colin Barker, Jul 22 2014

STATUS

approved