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%I #42 Jan 30 2024 04:38:35
%S 3,5,9,11,15,19,21,25,29,35,39,45,49,51,55,59,61,65,69,71,75,79,85,91,
%T 95,99,101,105,115,121,129,131,139,141,145,159,165,169,171,175,181,
%U 189,195,199,201,205,209,215,219,221
%N Numbers of the form y^2 - x^2 such that x^2 + y^2 is a prime and 0 < x < y.
%C What is the natural density of this set of these numbers?
%C 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
%C Numbers of the form s*t where 0 < s < t and (s^2 + t^2)/2 is prime. - _Robert Israel_, Jan 23 2017
%H Charles R Greathouse IV, <a href="/A281505/b281505.txt">Table of n, a(n) for n = 1..10000</a>
%H Sam Chow and Carl Pomerance, <a href="https://arxiv.org/abs/1703.10953">Triangles with prime hypotenuse</a>, arXiv:1703.10953 [math.NT], 2017.
%H Cihan Sabuncu, <a href="https://arxiv.org/abs/2401.16334">Right-angled triangles with almost prime hypotenuse</a>, arXiv:2401.16334 [math.NT], 2024. Mentions this sequence.
%F 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
%p filter:= proc(n)
%p ormap(s -> isprime((s^2 + (n/s)^2)/2), select(s -> s^2<n,
%p numtheory:-divisors(n)));
%p end proc:
%p select(filter, {seq(i,i=1..1000,2)}); # _Robert Israel_, Jan 23 2017
%t filter[n_] := AnyTrue[Select[Divisors[n], #^2 < n & ], PrimeQ[(#^2 + (n/#)^2)/2] & ];
%t Select[Range[1, 1000, 2], filter] (* _Jean-François Alcover_, Nov 27 2017, after _Robert Israel_ *)
%o (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
%Y Cf. A002144, A048161 is a subsequence, A070079 contains the same numbers.
%K nonn
%O 1,1
%A _Thomas Ordowski_, Jan 23 2017
%E More terms from _Altug Alkan_, Jan 23 2017
%E a(17)-a(50) from _Charles R Greathouse IV_, Jan 23 2017
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