A181419 Numbers of the form Fibonacci(p^{k+1})/Fibonacci(p^k) where p are primes, k>=1.

3, 7, 17, 47, 2207, 5777, 15005, 4870847, 598364773, 192900153617, 23725150497407, 792070839820228500005, 97415813466381445596089, 562882766124611619513723647, 400009475456580321242184872389193

Offset: 1

Author

Vladimir Shevelev, Oct 18 2010

Keywords

• nonn

Comments

The union of A001566 (p=2), A002814 except the first two terms (p=3), A145275 (p=5), A145277 (p=7), etc.

Links

• Robert Israel, Table of n, a(n) for n = 1..44

Crossrefs

Cf. A001566, A002814, A145275, A145277.

Programs

Maple

N:= 10^50: # for terms <= N
S:= {}: p:= 1:
do
 p:= nextprime(p);
 v:= combinat:-fibonacci(p);
 for k from 2 do
   w:= v;
   v:= combinat:-fibonacci(p^k);
   r:= v/w;
   if r > N then break fi;
   S:= S union {r};
 od;
 if k = 2 then break fi;
od:
sort(convert(S,list)); # Robert Israel, Apr 09 2024

Mathematica

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 *)

Extensions

a(11) onwards from Robert G. Wilson v, Oct 21 2010