A074691 -id:A074691 - cp's OEIS frontend

A061305 Squarefree Fibonacci numbers.

1, 1, 2, 3, 5, 13, 21, 34, 55, 89, 233, 377, 610, 987, 1597, 4181, 6765, 10946, 17711, 28657, 121393, 196418, 317811, 514229, 1346269, 2178309, 3524578, 5702887, 9227465, 24157817, 39088169, 63245986, 102334155, 165580141, 433494437, 701408733, 1134903170, 1836311903

Offset: 1

Amarnath Murthy, Apr 26 2001

Union of A074691 and A075735. - R. J. Mathar, Feb 06 2010
About 60% of entries are of the form 4k+1; 20% are of the form 4k+2; 20% are of the form 4k+3. Obviously no term is divisible by 4. - Carmine Suriano, Feb 27 2014
Contains A030426 as a subsequence unless there exist Wall-Sun-Sun primes. - Max Alekseyev, Jan 04 2018

			55 = 5 * 11 is a squarefree Fibonacci number.

Amiram Eldar, Table of n, a(n) for n = 1..1080 (terms 1..200 from Zak Seidov)

Intersection of A000045 and A005117.

a={}; Do[f=Fibonacci[n]; If[SquareFreeQ[f], AppendTo[a, f]], {n, 1, 50}]; a (* Vladimir Joseph Stephan Orlovsky, Jul 22 2008 *)
Select[Fibonacci[Range[50]],SquareFreeQ] (* Harvey P. Dale, Aug 26 2021 *)

{ n=0; g=0; f=1; for (i=1, 500, if (issquarefree(g), write("b061305.txt", n++, " ", g)); if (n==200, break); s=f; f+=g; g=s ) } \\ Harry J. Smith, Jul 21 2009

Set difference of A000045 and A061899. a(n) = A000045(A037918(n)). - R. J. Mathar, Feb 16 2010

More terms from Asher Auel, May 14 2001

Mathematica updated by Jean-François Alcover, Jul 04 2013

A075735 Squarefree Fibonacci numbers with an even number of prime factors (mu(n)=1).

Original entry on oeis.org

1, 1, 21, 34, 55, 377, 4181, 6765, 17711, 121393, 196418, 317811, 1346269, 2178309, 5702887, 102334155, 165580141, 32951280099, 53316291173, 139583862445, 956722026041, 2504730781961, 10610209857723, 308061521170129

Offset: 1

Jani Melik, Oct 07 2002

			21 is a Fibonacci number and 21=3*7, 34 is a Fibonacci numbers and 34=2*17, ...

Amiram Eldar, Table of n, a(n) for n = 1..505 (terms 1..100 from T. D. Noe)

Subsequence of A061305 (squarefree Fibonacci numbers).
Cf. A000045, A030229, A074691.
Cf. A022307, A053409, A072381.

with(combinat, fibonacci): m1_fib := proc(n); if (numtheory[mobius](fibonacci(n))=1) then RETURN(fibonacci(n)); fi; end: seq(m1_fib(i), i=1..100);

A075736 Fibonacci numbers F(k) as k runs through the products of an odd number of distinct primes A030059 (mu(k)=-1).

Original entry on oeis.org

1, 2, 5, 13, 89, 233, 1597, 4181, 28657, 514229, 832040, 1346269, 24157817, 165580141, 267914296, 433494437, 2971215073, 53316291173, 956722026041, 2504730781961, 27777890035288, 44945570212853, 190392490709135

Offset: 1

Jani Melik, Oct 07 2002

			30=2*3*5 and fibonacci(30)=832040, 31 is prime and fibonacci(31)=1346269,...

Cf. A000045, A030059, A074691.

A075736 := proc(n)
    combinat[fibonacci](A030059(n)) ;
end proc:
seq(A075736(n),n=1..40) ; # R. J. Mathar, Sep 22 2020

a(n) = A000045(A030059(n)).

A075742 Fibonacci numbers for which both the value and index are the product of an odd number of distinct primes; that is, numbers Fibonacci(k) for which mu(k) = mu(Fibonacci(k)) = -1.

Original entry on oeis.org

2, 5, 13, 89, 233, 1597, 28657, 514229, 24157817, 433494437, 2971215073, 44945570212853, 190392490709135, 99194853094755497, 1500520536206896083277, 3928413764606871165730, 1066340417491710595814572169, 19134702400093278081449423917

Offset: 1

Jani Melik, Oct 07 2002

			11 is a prime and Fibonacci(11) = 89 is a prime, 13 is a prime and Fibonacci(13) = 233 is a prime, but Fibonacci(16) = 987 = 3*7*47 and 16 is not squarefree and 30 = 2*3*5 is the product of an odd number of distinct primes but Fibonacci(30) = 832040 = 2^3*5*11*31*61 is not squarefree, ...

Cf. A000045, A008683, A030059, A074691.

with(combinat, fibonacci): m2_supM_fib := proc(n); if (numtheory[mobius](n)=-1) then if (numtheory[mobius](fibonacci(n))=-1) then RETURN(fibonacci(n)); fi; fi; end:

Incorrect 83621143489848422977 removed and more terms from Sean A. Irvine, Mar 05 2025

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A061305 Squarefree Fibonacci numbers.

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A075735 Squarefree Fibonacci numbers with an even number of prime factors (mu(n)=1).

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A075736 Fibonacci numbers F(k) as k runs through the products of an odd number of distinct primes A030059 (mu(k)=-1).

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A075742 Fibonacci numbers for which both the value and index are the product of an odd number of distinct primes; that is, numbers Fibonacci(k) for which mu(k) = mu(Fibonacci(k)) = -1.

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