A191375 Primes that are the sum of squares of three positive Fibonacci numbers.
3, 11, 17, 19, 43, 59, 137, 179, 347, 443, 449, 467, 491, 509, 569, 619, 883, 907, 1051, 1229, 1601, 2753, 3203, 3467, 3491, 3907, 6491, 8363, 8387, 8803, 20749, 20809, 21893, 24917, 28661, 41641, 44497, 49393, 54323, 55171, 62219, 75029, 108587, 284267, 372173
Offset: 1
Examples
43 = fib(3)^2 + fib(3)^2 + fib(5)^2 is prime.
Programs
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Mathematica
f = Union[Table[Fibonacci[n]^2, {n, 16}]]; t = Union[Flatten[Table[ f[[i]] + f[[j]] + f[[k]], {i, Length[f]}, {j, i, Length[f]}, {k, j, Length[f]}]]]; Select[t, # <= f[[-1]] && PrimeQ[#] &] (* T. D. Noe, Jun 03 2011 *) -
PARI
list(lim)=my(f=List(),v=List()); for(n=1,oo, my(t=fibonacci(n)^2); if(t+2>lim, break); listput(f,t)); for(i=1,#f, for(j=1,i, for(k=1,j, my(p=f[i]+f[j]+f[k]); if(p>lim, break); if(isprime(p), listput(v,p))))); Set(v) \\ Charles R Greathouse IV, Feb 24 2023
Formula
a(n) > exp(n^(1/3)*log(phi+1))/5. - Charles R Greathouse IV, Feb 24 2023