A037918 -id:A037918 - cp's OEIS frontend

A173208 Squarefree Fibonacci numbers F such that F+1 and F-1 are also squarefree.

2, 34, 610, 10946, 196418, 3524578, 63245986, 1134903170, 6557470319842, 117669030460994, 37889062373143906, 679891637638612258, 12200160415121876738, 3928413764606871165730, 1264937032042997393488322

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 12 2010

nonn

See A037918 for an implicit list of the squarefree F.

Chai Wah Wu, Table of n, a(n) for n = 1..49

Cf. A000045, A067259, A173207.

f[n_]:=Union[Last/@FactorInteger[n]][[ -1]];lst={};Do[fibo=Fibonacci[n]; If[f[fibo-1]==1&&f[fibo+1]==1&&f[fibo]==1,AppendTo[lst,fibo]],{n, 4,200}];lst
Select[Fibonacci[Range[200]],And@@SquareFreeQ/@{#-1,#,#+1}&] (* Harvey P. Dale, Nov 14 2011 *)

from sympy import factorint
A173208_list = [2]
a, b = 2, 3
for _ in range(10**2):
    if max(factorint(b).values()) <= 1 and max(factorint(b-1).values()) <= 1 and max(factorint(b+1).values()) <= 1:
        A173208_list.append(b)
    a, b = b, a + b # Chai Wah Wu, Jun 08 2015

Description simplified by R. J. Mathar, Feb 15 2010

Initial term 2 added, since 1 by convention is squarefree (see A005117) by Harvey P. Dale, Nov 15 2011

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).

A258575 Numbers n such that Lucas(n)-Fibonacci(n) is squarefree.

Original entry on oeis.org

0, 2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 53, 54, 56, 59, 60, 62, 63, 65, 66, 68, 69, 71, 72, 74, 75, 77, 78, 80, 81, 83, 84, 86, 87, 89, 90, 93, 95, 96, 98, 99, 102, 104, 105

Offset: 1

Vincenzo Librandi, Jun 04 2015

These numbers belong to the sequence A007494 (see Chai Wah Wu argumentation in A258574).
Also numbers n such that 2*Fibonacci(n-1) is squarefree. [Bruno Berselli, Jun 05 2015]
A258575(n) = A258574(n+1)-1. - Chai Wah Wu, Jun 09 2015

Cf. A000032, A000045, A005117, A007494, A037918, A118658, A258574.

[0] cat [n: n in [2..150] | IsSquarefree(Lucas(n)-Fibonacci(n))];

Select[Range[0, 200], SquareFreeQ[LucasL[#] - Fibonacci[#]] &]

[n for n in (0..110) if is_squarefree(2*fibonacci(n-1))] # Bruno Berselli, Jun 05 2015

Name corrected by Bruno Berselli, Jun 05 2015

A174279 Smallest k such that tau(Fibonacci(k)) = 2^n.

Original entry on oeis.org

1, 3, 6, 15, 18, 44, 30, 54, 128, 80, 138, 90, 162, 198, 308, 294, 210, 460, 288, 270, 378, 510, 680, 594, 920, 570, 690, 1280, 1190, 630, 1040, 1386, 810

Offset: 0

Michel Lagneau, Mar 15 2010

Smallest k such that A000005(A000045(k)) = 2^n.

The multiplicative property of the tau-function implies that the Fibonacci(k) has a prime factor representation p_1^e_1*p_2^e_2*... where (e_1+1)*(e_2+1)*... is a power of 2, that is, the exponents are in {1,3,7,15,...}. This adds for example the squarefree Fibonacci numbers with indices from A037918 to the list of candidates. - R. J. Mathar, Oct 11 2011

			a(0) =  1 because tau(Fibonacci(1))  = tau(1)   = 2^0 = 1.
a(1) =  3 because tau(Fibonacci(3))  = tau(2)   = 2^1 = 2.
a(2) =  6 because tau(Fibonacci(6))  = tau(8)   = 2^2 = 4.
a(3) = 15 because tau(Fibonacci(15)) = tau(610) = 2^3 = 8.

Majorie Bicknell and Verner E Hoggatt, Fibonacci's Problem Book, Fibonacci Association, San Jose, Calif., 1974.

Cf. A085077, A063375.

with(numtheory):for p from 1 to 100 do:indic:=0:u0:=0:u1:=1:for n from 2 to 1000 while(indic=0)do:s:=u0+u1:u0:=u1:u1:=s:if tau(s)= 2^p and indic=0 then print(p): print(n): indic:=1:else fi:od:od:

a(27)-a(32) from Amiram Eldar, Oct 14 2019

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A173208 Squarefree Fibonacci numbers F such that F+1 and F-1 are also squarefree.

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A250093 Squarefree part of Fibonacci(n^2).

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A258575 Numbers n such that Lucas(n)-Fibonacci(n) is squarefree.

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A174279 Smallest k such that tau(Fibonacci(k)) = 2^n.

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