Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #31 Oct 14 2019 03:33:21
%S 1,2,5,8,11,12,25,29,32,41,48,55,89,121,125,128,131,145,179,192,205,
%T 275,331,359,401,421,431,445,449,509,512,569,571,601,605,625,631,655,
%U 659,691,725,768,895,911,1025,1375
%N Numbers n such that the Fibonacci number F(n) can be written in the form a^2 + 5*b^2.
%C A number n can be written in the form a^2+5*b^2 if and only if n is 0, 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) 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), for integers i,j,k,m, for primes p,q.
%H Blair Kelly, <a href="http://mersennus.net/fibonacci">Fibonacci and Lucas factorizations</a>
%o (PARI) for(i=2, 500, a=factorint(fibonacci(i))~; 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", ")))
%Y Cf. A000045, A215939, A124132.
%Y Cf. A020669, A033205 (numbers and primes of the form x^2 + 5*y^2).
%K nonn,more
%O 1,2
%A _V. Raman_, Aug 27 2012
%E Terms corrected by _V. Raman_, Sep 20 2012
%E a(46) from _Amiram Eldar_, Oct 14 2019