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Numbers k such that prime(k) is the hypotenuse of a Pythagorean triple where one leg is also prime.
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%I #35 Apr 04 2021 00:53:25

%S 3,6,18,42,82,271,284,369,445,682,1069,1193,1900,2241,3894,6137,7108,

%T 8164,9658,10126,12645,14842,14936,17913,18420,19480,23893,24605,

%U 28959,32913,36279,40847,43936,44559,45500

%N Numbers k such that prime(k) is the hypotenuse of a Pythagorean triple where one leg is also prime.

%C In such a triangle, the leg that is not prime is always the largest one and is equal to prime(k)-1; these even legs are in A067755. E.g. for a(2) = 6, prime(6) = 13 and the corresponding Pythagorean triple is (5, 12, 13). - _Bernard Schott_, Apr 03 2021

%H Robert Israel, <a href="/A342583/b342583.txt">Table of n, a(n) for n = 1..1000</a>

%e a(1) = 3, since prime(3) = 5 is the hypotenuse of the triple (3,4,5).

%p R:= NULL: count:= 0:

%p p:= 2:

%p while count < 100 do

%p p:= nextprime(p); n:= (p-1)/2; q:= 2*n^2+2*n+1;

%p if isprime(q) then

%p count:= count+1; r:= numtheory:-pi(q); R:= R, r;

%p fi

%p od:

%p R; # _Robert Israel_, Mar 22 2021

%t PrimePi[Take[Cases[Import["https://oeis.org/A067756/b067756.txt","Table"],{_,_}][[All,2]],100]]

%Y Cf. A067756 (the hypotenuses).

%Y Cf. A009000, A046083, A046084, A067755.

%K nonn

%O 1,1

%A _Ivan N. Ianakiev_, Mar 16 2021