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A371866
Primes of the form Fibonacci(m^(k+1))/Fibonacci(m^k), where m > 1 and k >= 1.
0
3, 7, 17, 47, 2207, 97415813466381445596089
OFFSET
1,1
COMMENTS
a(7) > 10^25000 if it exists.
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).
EXAMPLE
a(1) = 3 = F(2^2)/F(2^1) where F = Fibonacci.
a(2) = 7 = F(2^3)/F(2^2).
a(3) = 17 = F(3^2)/F(3^1).
a(4) = 47 = F(2^4)/F(2^3).
a(5) = 2207 = F(2^5)/F(2^4).
a(6) = 97415813466381445596089 = F(11^2)/F(11^1).
MAPLE
N:= 10^1000: # for terms < N
R:= NULL: F:= combinat:-fibonacci:
p:= 1:
do
p:= nextprime(p);
v:= F(p);
for k from 2 do
w:= v;
v:= F(p^k);
r:= v/w;
if r > N then break fi;
if isprime(r) then R:= R, r fi;
od;
if k = 2 then break fi;
od:
sort([R]);
CROSSREFS
Primes in A181419.
Cf. A000045.
Sequence in context: A179944 A071985 A181419 * A090977 A324789 A014144
KEYWORD
nonn
AUTHOR
Robert Israel, Apr 09 2024
STATUS
approved