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A371866 Primes of the form Fibonacci(m^(k+1))/Fibonacci(m^k), where m > 1 and k >= 1.

%I A371866 #13 Mar 29 2025 20:07:35
%S A371866 3,7,17,47,2207,97415813466381445596089
%N A371866 Primes of the form Fibonacci(m^(k+1))/Fibonacci(m^k), where m > 1 and k >= 1.
%C A371866 a(7) > 10^25000 if it exists.
%C A371866 m must be prime, as Fibonacci((a*b)^(k+1))/Fibonacci((a*b)^k) = (Fibonacci((a*b)^(k+1))/Fibonacci(a^k * b^(k+1))) * Fibonacci(a^k * b^(k+1))/Fibonacci((a*b)^k).
%e A371866 a(1) = 3 = F(2^2)/F(2^1) where F = Fibonacci.
%e A371866 a(2) = 7 = F(2^3)/F(2^2).
%e A371866 a(3) = 17 = F(3^2)/F(3^1).
%e A371866 a(4) = 47 = F(2^4)/F(2^3).
%e A371866 a(5) = 2207 = F(2^5)/F(2^4).
%e A371866 a(6) = 97415813466381445596089 = F(11^2)/F(11^1).
%p A371866 N:= 10^1000: # for terms < N
%p A371866 R:= NULL: F:= combinat:-fibonacci:
%p A371866 p:= 1:
%p A371866 do
%p A371866  p:= nextprime(p);
%p A371866  v:= F(p);
%p A371866  for k from 2 do
%p A371866    w:= v;
%p A371866    v:= F(p^k);
%p A371866    r:= v/w;
%p A371866    if r > N then break fi;
%p A371866    if isprime(r) then R:= R, r fi;
%p A371866  od;
%p A371866  if k = 2 then break fi;
%p A371866 od:
%p A371866 sort([R]);
%Y A371866 Primes in A181419.
%Y A371866 Cf. A000045.
%K A371866 nonn
%O A371866 1,1
%A A371866 _Robert Israel_, Apr 09 2024