A236264 - OEIS

%I #38 Jun 28 2022 01:48:18

%S 0,2,6,12,14,26,38,62,74,86,98,122,134,146,158,182,222,254,326,338,

%T 366,398,446,614,626,698,722,794,866,1022,1046,1082,1226,1238,1418,

%U 1646,1814,2174,2246,2258,2294,2426,2558

%N Even indices of Fibonacci numbers which are the sum of two squares.

%C The first 10 such Fibonacci numbers are 0, 1, 8, 144, 377, 121393, 39088169, 4052739537881, 1304969544928657, 420196140727489673.

%C Ballot & Luca (Proposition 1) show that this sequence has asymptotic density 0. - _Charles R Greathouse IV_, Jan 21 2014

%C a(43) >= 2558. Determining this term requires factoring the Lucas number L_1279. - _Charles R Greathouse IV_, Jan 21 2014

%C 3002 <= a(44) <= 3302. 3302, 3698, 4898 are terms. - _Chai Wah Wu_, Jul 23 2020

%H Christian Ballot and Florian Luca, <a href="http://dx.doi.org/10.4064/aa127-2-4">On the equation x^2+dy^2=Fn</a>, Acta Arith. 127 (2007), 145-155.

%H Kevin O'Bryant, <a href="http://mathoverflow.net/questions/67601">Which Fibonacci numbers are the sum of two squares?</a>, MathOverflow.

%F a(n) = 2*A124132(n-1).

%e Fibonacci(14) = 377 = 19^2 + 4^2, so 14 is in the sequence.

%t Reap[For[n = 0, n <= 400, n = n+2, If[Reduce[Fibonacci[n] == x^2 + y^2, {x, y}, Integers] =!= False, Print[n]; Sow[n]]]][[2, 1]]

%o (PARI) is(n)=if(n%2, return(0)); my(f=factor(fibonacci(n))); for(i=1,#f~, if(f[i,1]%4==3 && f[i,2]%2, return(0))); 1 \\ _Charles R Greathouse IV_, Jan 21 2014

%o (PARI) default(factor_add_primes, 1);

%o is(n)={

%o     if(n%2,return(0));

%o     my(f=fibonacci(n),t);

%o     if(f%4==3,return(0));

%o     forprime(p=2,min(log(f)^2,1e5),

%o         if(f%p==0,

%o             t=valuation(f,p);

%o             if(p%4==3&&t%2,return(0));

%o             f/=p^t;

%o             if(f%4==3,return(0))

%o         )

%o     );

%o     fordiv(n,d,

%o         if(d==n, break);

%o         t=factor(fibonacci(d))[,1];

%o         for(i=1,#t,

%o             if(t[i]%4==3 && valuation(f,t[i])%2, return(0));

%o             f/=t[i]^valuation(f,t[i]);

%o             if(f%4==3,return(0))

%o         )

%o     );

%o     f=factor(f);

%o     for(i=1,#f[,1],

%o         if(f[i,2]%2&&f[i,1]%4==3,return(0))

%o     );

%o     1

%o }; \\ _Charles R Greathouse IV_, Jan 21 2014

%o (Python)

%o from itertools import count, islice

%o from sympy import factorint, fibonacci

%o def A236264_gen(): # generator of terms

%o     return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(fibonacci(n)).items()),count(0,2))

%o a236264_list = list(islice(A236264_gen(),10)) # _Chai Wah Wu_, Jun 27 2022

%Y Cf. A000045, A001481, A124132.

%K nonn,more

%O 1,2

%A _Jean-François Alcover_, Jan 21 2014

%E a(32)-a(42) from _Charles R Greathouse IV_, Jan 21 2014

%E a(43) from _Chai Wah Wu_, Jul 23 2020