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%I #29 Sep 08 2022 08:45:51
%S 1,2,3,4,5,6,7,11,13,17,20,23,24,27,28,29,32,43,47,52,55,74,77,80,83,
%T 84,85,87,88,91,93,96,97,100,108,115,123,131,132,137,138,143,146,149,
%U 156,157,161,163,178,184,187,189,196,197,209,211,214,215,221,222,223,232
%N Numbers n such that omega(Fibonacci(n)) is a square.
%C 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.
%D Majorie Bicknell and Verner E Hoggatt, Fibonacci's Problem Book, Fibonacci Association, San Jose, Calif., 1974.
%D Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, The Fibonacci Association, 1972, pages 1-8.
%H Amiram Eldar, <a href="/A174323/b174323.txt">Table of n, a(n) for n = 1..258</a> (terms 1..200 from Robert Israel, derived from b-file for A022307)
%H Blair Kelly, <a href="http://mersennus.net/fibonacci//">Fibonacci and Lucas Factorizations</a>
%H Pieter Moree, <a href="http://msp.org/pjm/1998/186-2/p03.xhtml">Counting Divisors of Lucas Numbers</a>, Pacific J. Math, Vol. 186, No. 2, 1998, pp. 267-284.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fibonacci_number">Fibonacci number</a>
%e omega(Fibonacci(1)) = omega(Fibonacci(2)) = omega(1) = 0,
%e omega(Fibonacci(3)) = omega(2) = 1,
%e omega(Fibonacci(20)) = omega(6765)= 4,
%e omega(Fibonacci(80)) = omega( 23416728348467685) = 9.
%p 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:
%p # alternative:
%p P[1]:= {}: count:= 1: res:= 1:
%p for i from 2 to 300 do
%p pn:= map(t -> i/t, numtheory:-factorset(i));
%p Cprimes:= `union`(seq(P[t],t=pn));
%p f:= combinat:-fibonacci(i);
%p for p in Cprimes do f:= f/p^padic:-ordp(f,p) od;
%p P[i]:= Cprimes union numtheory:-factorset(f);
%p if issqr(nops(P[i])) then
%p count:= count+1;
%p res:= res, i;
%p fi;
%p od:
%p res; # _Robert Israel_, Oct 13 2016
%t Select[Range[200], IntegerQ[Sqrt[PrimeNu[Fibonacci[#]]]] &] (* _G. C. Greubel_, May 16 2017 *)
%o (PARI) is(n)=issquare(omega(fibonacci(n))) \\ _Charles R Greathouse IV_, Oct 13 2016
%o (Magma) [k:k in [1..240]| IsSquare(#PrimeDivisors(Fibonacci(k)))]; // _Marius A. Burtea_, Oct 15 2019
%Y Cf. A038575 Number of prime factors of n-th Fibonacci number, with multiplicity.
%Y Cf. A000213, A000288, A000322, A000383, A060455, A030186, A039834, A020695, A020701, A071679.
%Y Cf. A022307 Number of distinct prime factors of n-th Fibonacci number A086597 (number of primitive prime factors).
%K nonn
%O 1,2
%A _Michel Lagneau_, Mar 15 2010
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