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%I #52 Jul 23 2020 02:39:48
%S 3,7,13,19,31,37,43,61,67,73,79,127,163,199,223,307,313,349,397,433,
%T 523,541,613,619,709,823,907,1087,1123,1129,1213,1279,1531
%N Prime numbers n for which the Lucas number L(n) (see A000032) is the sum of two squares.
%C These Lucas numbers L(n) have no prime factor congruent to 3 mod 4 to an odd power.
%C Also prime numbers n such that the Lucas number L(n) can be written in the form a^2 + 5*b^2.
%C Any prime factor of Lucas(n) for n prime is always of the form 1 (mod 10) or 9 (mod 10).
%C A number n can be written in the form a^2+5*b^2 (see A020669) if and only if n is 0,
%C or of the form 2^(2i) 5^j Prod_{p==1 or 9 mod 20} p^k Prod_{q==3 or 7 mod 20) q^(2m)
%C or of the form 2^(2i+1) 5^j Prod_{p==1 or 9 mod 20} p^k Prod_{q==3 or 7 mod 20) q^(2m+1),
%C for integers i,j,k,m, for primes p,q.
%C 1607 <= a(34) <= 1747. 1747, 1951, 2053, 2467, 5107, 5419, 5851, 7243, 7741, 8467, 13963, 14449, 14887, 15511, 15907, 35449, 51169, 193201, 344293, 387433, 574219, 901657, 1051849 are terms. - _Chai Wah Wu_, Jul 22 2020
%H Blair Kelly, <a href="http://mersennus.net/fibonacci">Fibonacci and Lucas factorizations</a>
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/SumofSquaresFunction.html">Sum of squares function</a>
%e Lucas(19) = 9349 = 95^2 + 18^2.
%e Lucas(19) = 9349 = 23^2 + 5*42^2.
%o (PARI) forprime(i=2, 500, a=factorint(fibonacci(i-1)+fibonacci(i+1))~; has=0; for(j=1, #a, if(a[1, j]%4==3&&a[2, j]%2==1, has=1; break)); if(has==0, print(i", "))) \\ a^2+b^2 form.
%o (PARI) forprime(i=2, 500, a=factorint(fibonacci(i-1)+fibonacci(i+1))~; flag=0; flip=0; for(j=1, #a, if(((a[1, j]%20>10))&&a[2, j]%2==1, flag=1); if(((a[1, j]%20==2)||(a[1, j]%20==3)||(a[1, j]%20==7))&&a[2, j]%2==1, flip=flip+1)); if(flag==0&&flip%2==0, print(i", "))) \\ a^2+5*b^2 form.
%Y Cf. A000032, A001481, A124130, A180363, A215906, A215907.
%Y Cf. A020669, A033205 (numbers and primes of the form x^2 + 5*y^2).
%K nonn
%O 1,1
%A _V. Raman_, Aug 23 2012
%E Merged A215941 into this sequence, _T. D. Noe_, Sep 21 2012
%E a(30)-a(33) from _Chai Wah Wu_, Jul 22 2020
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