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A347163 Primes p such that 10*p can be written as a^2+b^2 where a and b are prime. 1


%S 5,13,17,29,37,41,53,89,97,101,113,137,149,173,197,233,257,269,281,

%T 293,317,349,353,373,389,401,449,461,509,521,557,569,593,617,641,653,

%U 677,701,761,773,797,809,821,857,881,929,953,977,1013,1021,1049,1061,1097,1109,1181,1193,1217,1229,1289

%N Primes p such that 10*p can be written as a^2+b^2 where a and b are prime.

%C Pythagorean primes p = x^2+y^2 where 3*x+y and |x-3*y| or x+3*y and |3*x-y| are primes.

%H Robert Israel, <a href="/A347163/b347163.txt">Table of n, a(n) for n = 1..10000</a>

%e a(3) = 17 is a term because 170 = 7^2+11^2 with 17, 7 and 11 all prime.

%p filter:= proc(p) local F,a,b; uses GaussInt;

%p if not isprime(p) then return false fi;

%p F:= GIfactors(p)[2][1][1];

%p a:= abs(Re(F)); b:= abs(Im(F));

%p (isprime(a+3*b) and isprime(abs(3*a-b))) or (isprime(3*a+b) and isprime(abs(3*b-a)))

%p end proc:

%p select(filter, [seq(i,i=5..10000,4)]);

%t Select[Prime@Range@300,Select[PowersRepresentations[10#,2,2],And@@PrimeQ@#&]!={}&] (* _Giorgos Kalogeropoulos_, Aug 20 2021 *)

%Y Contained in A002144.

%K nonn

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Aug 20 2021

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Last modified October 1 04:31 EDT 2022. Contains 357134 sequences. (Running on oeis4.)