%I #14 Apr 10 2024 03:38:06
%S 3,7,17,47,2207,5777,15005,4870847,598364773,192900153617,
%T 23725150497407,792070839820228500005,97415813466381445596089,
%U 562882766124611619513723647,400009475456580321242184872389193
%N Numbers of the form Fibonacci(p^{k+1})/Fibonacci(p^k) where p are primes, k>=1.
%C The union of A001566 (p=2), A002814 except the first two terms (p=3), A145275 (p=5), A145277 (p=7), etc.
%H Robert Israel, <a href="/A181419/b181419.txt">Table of n, a(n) for n = 1..44</a>
%p N:= 10^50: # for terms <= N
%p S:= {}: p:= 1:
%p do
%p p:= nextprime(p);
%p v:= combinat:-fibonacci(p);
%p for k from 2 do
%p w:= v;
%p v:= combinat:-fibonacci(p^k);
%p r:= v/w;
%p if r > N then break fi;
%p S:= S union {r};
%p od;
%p if k = 2 then break fi;
%p od:
%p sort(convert(S,list)); # _Robert Israel_, Apr 09 2024
%t t = Sort@ Flatten[ Table[ {Prime[n]^(e + 1), Prime[n]^e}, {n, 8}, {e, 10}], 1]; u = Select[t, First@# < 350 &]; Sort[ Fibonacci[ #[[1]]]/Fibonacci[ #[[2]]] & /@ u] (* _Robert G. Wilson v_, Oct 21 2010 *)
%Y Cf. A001566, A002814, A145275, A145277.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, Oct 18 2010
%E a(11) onwards from _Robert G. Wilson v_, Oct 21 2010