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A174323 Numbers n such that omega(Fibonacci(n)) is a square.

%I A174323 #36 May 07 2025 12:20:30
%S A174323 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 A174323 84,85,87,88,91,93,96,97,100,108,115,123,131,132,137,138,143,146,149,
%U A174323 156,157,161,163,178,184,187,189,196,197,209,211,214,215,221,222,223,232
%N A174323 Numbers n such that omega(Fibonacci(n)) is a square.
%C A174323 Numbers n such that omega(A000045(n)) is a square, where omega(p) is the number of distinct prime 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 prime factors.
%D A174323 Majorie Bicknell and Verner E Hoggatt, Fibonacci's Problem Book, Fibonacci Association, San Jose, Calif., 1974.
%D A174323 Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, The Fibonacci Association, 1972, pages 1-8.
%H A174323 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 A174323 Blair Kelly, <a href="http://mersennus.net/fibonacci//">Fibonacci and Lucas Factorizations</a>
%H A174323 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 A174323 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>
%H A174323 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fibonacci_number">Fibonacci number</a>
%e A174323 omega(Fibonacci(1)) = omega(Fibonacci(2)) = omega(1) = 0,
%e A174323 omega(Fibonacci(3)) = omega(2) = 1,
%e A174323 omega(Fibonacci(20)) = omega(6765) = 4,
%e A174323 omega(Fibonacci(80)) = omega(23416728348467685) = 9.
%p A174323 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 A174323 # alternative:
%p A174323 P[1]:= {}: count:= 1: res:= 1:
%p A174323 for i from 2 to 300 do
%p A174323   pn:= map(t -> i/t, numtheory:-factorset(i));
%p A174323   Cprimes:= `union`(seq(P[t],t=pn));
%p A174323   f:= combinat:-fibonacci(i);
%p A174323   for p in Cprimes do f:= f/p^padic:-ordp(f,p) od;
%p A174323   P[i]:= Cprimes union numtheory:-factorset(f);
%p A174323   if issqr(nops(P[i])) then
%p A174323      count:= count+1;
%p A174323      res:= res, i;
%p A174323   fi;
%p A174323 od:
%p A174323 res; # _Robert Israel_, Oct 13 2016
%t A174323 Select[Range[200], IntegerQ[Sqrt[PrimeNu[Fibonacci[#]]]] &] (* _G. C. Greubel_, May 16 2017 *)
%o A174323 (PARI) is(n)=issquare(omega(fibonacci(n))) \\ _Charles R Greathouse IV_, Oct 13 2016
%o A174323 (Magma) [k:k in [1..240]| IsSquare(#PrimeDivisors(Fibonacci(k)))]; // _Marius A. Burtea_, Oct 15 2019
%Y A174323 Cf. A038575 (number of prime factors of n-th Fibonacci number, with multiplicity).
%Y A174323 Cf. A000045, A000213, A000288, A000322, A000383, A001221, A060455, A030186, A039834, A020695, A020701, A071679.
%Y A174323 Cf. A022307 (number of distinct prime factors of n-th Fibonacci number), A086597 (number of primitive prime factors).
%K A174323 nonn
%O A174323 1,2
%A A174323 _Michel Lagneau_, Mar 15 2010