A281505 Numbers of the form y^2 - x^2 such that x^2 + y^2 is a prime and 0 < x < y.

3, 5, 9, 11, 15, 19, 21, 25, 29, 35, 39, 45, 49, 51, 55, 59, 61, 65, 69, 71, 75, 79, 85, 91, 95, 99, 101, 105, 115, 121, 129, 131, 139, 141, 145, 159, 165, 169, 171, 175, 181, 189, 195, 199, 201, 205, 209, 215, 219, 221

Offset: 1

Author

Thomas Ordowski, Jan 23 2017

Keywords

• nonn

Comments

What is the natural density of this set of these numbers?

There are 204 terms up to 10^3, 1849 up to 10^4, 16881 up to 10^5, 160194 up to 10^6, 1531730 up to 10^7, and 14766494 up to 10^8. - Charles R Greathouse IV, Jan 23 2017

Numbers of the form s*t where 0 < s < t and (s^2 + t^2)/2 is prime. - Robert Israel, Jan 23 2017

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Sam Chow and Carl Pomerance, Triangles with prime hypotenuse, arXiv:1703.10953 [math.NT], 2017.
• Cihan Sabuncu, Right-angled triangles with almost prime hypotenuse, arXiv:2401.16334 [math.NT], 2024. Mentions this sequence.

Crossrefs

Cf. A002144, A048161 is a subsequence, A070079 contains the same numbers.

Programs

Maple

filter:= proc(n)
  ormap(s -> isprime((s^2 + (n/s)^2)/2), select(s -> s^2Robert Israel, Jan 23 2017

Mathematica

filter[n_] := AnyTrue[Select[Divisors[n], #^2 < n & ], PrimeQ[(#^2 + (n/#)^2)/2] & ];
Select[Range[1, 1000, 2], filter] (* Jean-François Alcover, Nov 27 2017, after Robert Israel *)

PARI

list(lim)=my(v=List()); for(a=1,sqrtint(lim\=1), for(x=1,(lim-a^2)\2\a, if(isprime((x+a)^2+x^2), listput(v,(x+a)^2-x^2)))); Set(v) \\ Charles R Greathouse IV, Jan 23 2017

Formula

a(n) = n(log n)^c /(log log n)^O(1), where c = 1 - (1 + log log 2)/log 2 = 0.086... Cf. A027424. - Conjectured by Carl Pomerance, Jan 25 2017

Extensions

More terms from Altug Alkan, Jan 23 2017

a(17)-a(50) from Charles R Greathouse IV, Jan 23 2017