A061899 -id:A061899 - 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

A235383 Fibonacci numbers that are the product of other Fibonacci numbers.

Original entry on oeis.org

8, 144

Offset: 1

Robert C. Lyons, Jan 08 2014

This sequence and A229037 and A235265 are winners in the contest held at the 2014 AMS/MAA Joint Mathematics Meetings. - T. D. Noe, Jan 20 2014
Carmichael's theorem implies that 8 and 144 are the only terms of this sequence.
First two terms of A061899, A111687, A172150, A212703, and A231851. - Omar E. Pol, Jan 21 2014
Saha and Karthik conjectured (without reference to Carmichael's theorem) that the only positive integers k for which A001175(k^2) = A001175(k) are 6 and 12. (A000045(6) = 8 and A000045(12) = 144.) - L. Edson Jeffery, Feb 13 2014
Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only nontrivial perfect power Fibonacci numbers. - Robert C. Lyons, Dec 23 2015

			The Fibonacci number 8 is in the sequence because 8=2*2*2, and 2 is a Fibonacci number that is not equal to 8. The Fibonacci number 144 is in the sequence because 144=3*3*2*2*2*2, and both 2 and 3 are Fibonacci numbers that are not equal to 144.

Yann Bugeaud, Maurice Mignotte, and Samir Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Annals of Mathematics, 163 (2006), pp. 969-1018.
Arpan Saha and Karthik C S, A few equivalences of Wall-Sun-Sun prime conjecture, arXiv:1102.1636 [math.NT], 2011.
Wikipedia, Carmichael's theorem.

Cf. A000045, A061899, A065108, A227875.

A173207 Cubefree Fibonacci numbers which are not squarefree.

Original entry on oeis.org

75025, 12586269025, 225851433717, 2111485077978050, 4660046610375530309, 354224848179261915075, 43566776258854844738105, 114059301025943970552219, 42230279526998466217810220532898, 1672445759041379840132227567949787325

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 12 2010

nonn

Subsequence of A061899. [R. J. Mathar, Feb 16 2010]

Apparently contains Fibonacci(25*m), m>0. - Ralf Stephan, Jan 26 2014

Chai Wah Wu, Table of n, a(n) for n = 1..21

f[n_]:=Union[Last/@FactorInteger[n]][[ -1]];lst={};Do[fibo=Fibonacci[n];If[f[fibo]==2,AppendTo[lst,fibo]],{n,3,2*5!}];lst
Select[Fibonacci[Range[200]],!SquareFreeQ[#]&&Max[FactorInteger[#][[;;,2]]]<3&] (* Harvey P. Dale, Nov 24 2024 *)

from sympy import factorint
A173207_list = []
a, b = 1, 2
for _ in range(10**2):
    if max(factorint(b).values()) == 2:
        A173207_list.append(b)
    a, b = b, a + b # Chai Wah Wu, Jun 08 2015

A000045 INTERSECT A067259.

Converted cross-references to a formula - R. J. Mathar, Feb 16 2010

A075739 Nonsquarefree Fibonacci numbers whose indices are also not squarefree.

Original entry on oeis.org

144, 2584, 46368, 75025, 14930352, 4807526976, 12586269025, 86267571272, 225851433717, 1548008755920, 498454011879264, 2111485077978050, 160500643816367088, 2880067194370816120, 51680708854858323072, 354224848179261915075, 16641027750620563662096

Offset: 1

Jani Melik, Oct 07 2002

			Fib(18)=2584 is there because neither 18 nor 2584 is squarefree.

Cf. A000045, A013929, A061899, A075738.

with(combinat, fibonacci): fib_supM_ndsk := proc(n); if (numtheory[issqrfree](n)='false') then if (numtheory[issqrfree](fibonacci(n))='false') then RETURN(fibonacci(n)); fi; fi; end: seq(fib_supM_ndsk(i), i=1..200);

Select[Table[{n,Fibonacci[n]},{n,300}],NoneTrue[#,SquareFreeQ]&][[All,2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 20 2019 *)

Edited by Don Reble, Nov 05 2005

More terms from Harvey P. Dale, Aug 20 2019

A250093 Squarefree part of Fibonacci(n^2).

Original entry on oeis.org

1, 3, 34, 987, 3001, 103683, 7778742049, 10610209857723, 37889062373143906, 14168993927170476603, 8670007398507948658051921, 964523271222730372229194083, 93202207781383214849429075266681969, 40934782466626840596168752972961528246147

Offset: 1

Vincenzo Librandi, Nov 12 2014

nonn

Also, the smallest number such that a(n)*Fibonacci(n^2) is a square.

Conjecture: the only primes in this sequence are 3 and 3001.

Amiram Eldar, Table of n, a(n) for n = 1..37

Cf. A037918, A061305, A061899, A063509, A069110.

[SquareFree(Fibonacci(n^2)): n in [1..20]];

Table[Times@@Power@@@({#[[1]], Mod[#[[2]], 2]}&/@FactorInteger[Fibonacci[n^2]]), {n, 20}]

for(n=1, 60, print1(core(fibonacci(n^2)), ", "))

a(n) = A069110(n^2).

A075732 Fibonacci numbers F(k) for k not squarefree (A013929).

Original entry on oeis.org

3, 21, 34, 144, 987, 2584, 6765, 46368, 75025, 196418, 317811, 2178309, 14930352, 102334155, 701408733, 1134903170, 4807526976, 7778742049, 12586269025, 32951280099, 86267571272, 225851433717, 1548008755920, 6557470319842

Offset: 0

Jani Melik, Oct 07 2002

			4 is a nonsquarefree number and Fibonacci(4)=3, 8 is a nonsquarefree number and Fibonacci(8)=21,...

Cf. A000045, A013929, A061899.

with(combinat, fibonacci): fib_ndsk := proc(n); if (numtheory[issqrfree](n)='false') then # or # if (mobius(n)=0) then RETURN(fibonacci(n)) fi; end: seq(fib_ndsk(i), i=1..100);

Fibonacci[#]&/@DeleteCases[Range[100],?SquareFreeQ]  (* _Harvey P. Dale, Apr 20 2011 *)

cp's OEIS Frontend

A061305 Squarefree Fibonacci numbers.

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A235383 Fibonacci numbers that are the product of other Fibonacci numbers.

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A173207 Cubefree Fibonacci numbers which are not squarefree.

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A075739 Nonsquarefree Fibonacci numbers whose indices are also not squarefree.

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A250093 Squarefree part of Fibonacci(n^2).

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A075732 Fibonacci numbers F(k) for k not squarefree (A013929).

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