OFFSET
1,2
COMMENTS
Numbers n such that omega(A000045 (n)) is a square, where omega(p) is the number of distinct primes factors of p (A001221). Remark: for the larger Fibonacci numbers F(n) (n > 300), the Maple program (below) is very slow. So we use a two-step process: factoring F(n) with the elliptic curve method, and then calculate the distinct primes factors.
REFERENCES
Majorie Bicknell and Verner E Hoggatt, Fibonacci's Problem Book, Fibonacci Association, San Jose, Calif., 1974.
Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, The Fibonacci Association, 1972, pages 1-8.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..258 (terms 1..200 from Robert Israel, derived from b-file for A022307)
Blair Kelly, Fibonacci and Lucas Factorizations
Pieter Moree, Counting Divisors of Lucas Numbers, Pacific J. Math, Vol. 186, No. 2, 1998, pp. 267-284.
Eric Weisstein's World of Mathematics, Fibonacci Number
Wikipedia, Fibonacci number
EXAMPLE
omega(Fibonacci(1)) = omega(Fibonacci(2)) = omega(1) = 0,
omega(Fibonacci(3)) = omega(2) = 1,
omega(Fibonacci(20)) = omega(6765)= 4,
omega(Fibonacci(80)) = omega( 23416728348467685) = 9.
MAPLE
with(numtheory):u0:=0:u1:=1:for p from 2 to 400 do :s:=u0+u1:u0:=u1:u1:=s: s1:=nops( ifactors(s)[2]): w1:=sqrt(s1):w2:=floor(w1):if w1=w2 then print (p): else fi:od:
# alternative:
P[1]:= {}: count:= 1: res:= 1:
for i from 2 to 300 do
pn:= map(t -> i/t, numtheory:-factorset(i));
Cprimes:= `union`(seq(P[t], t=pn));
f:= combinat:-fibonacci(i);
for p in Cprimes do f:= f/p^padic:-ordp(f, p) od;
P[i]:= Cprimes union numtheory:-factorset(f);
if issqr(nops(P[i])) then
count:= count+1;
res:= res, i;
fi;
od:
res; # Robert Israel, Oct 13 2016
MATHEMATICA
Select[Range[200], IntegerQ[Sqrt[PrimeNu[Fibonacci[#]]]] &] (* G. C. Greubel, May 16 2017 *)
PROG
(PARI) is(n)=issquare(omega(fibonacci(n))) \\ Charles R Greathouse IV, Oct 13 2016
(Magma) [k:k in [1..240]| IsSquare(#PrimeDivisors(Fibonacci(k)))]; // Marius A. Burtea, Oct 15 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Mar 15 2010
STATUS
approved