A037917 -id:A037917 - cp's OEIS frontend

A037918 Numbers n such that the Fibonacci number F(n) is squarefree.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 76, 77, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 92

Offset: 1

N. J. A. Sloane, Marc LeBrun

nonn

			10 is in the sequence because F(10) = 55, which is squarefree since 55 = 5 * 11.
11 is in the sequence because F(11) = 89, which is prime and therefore also squarefree.
12 is not in the sequence because F(12) = 144 = 2^4 * 3^2 = 12^2.

Amiram Eldar, Table of n, a(n) for n = 1..1080 (terms 1..765 from T. D. Noe, based on Kelly's table)
Blair Kelly, Fibonacci Factorizations
Eric Weisstein's World of Mathematics, Fibonacci Number.

Cf. A000045, A037917.

[n: n in [1..100] | IsSquarefree(Fibonacci(n))]; // Vincenzo Librandi, Jun 01 2015

Select[Range[100], SquareFreeQ[Fibonacci[#]] &] (* Vladimir Joseph Stephan Orlovsky, Jul 01 2011 *)
Select[Range[100], MoebiusMu[Fibonacci[#]] != 0 &] (* Alonso del Arte, Jan 26 2014 *)

select(n->(0!=moebius(fibonacci(n))),vector(100,j,j)) \\ Joerg Arndt, Jan 28 2014

More terms from James Sellers, May 29 2000

A061899 Fibonacci numbers that are not squarefree.

Original entry on oeis.org

8, 144, 2584, 46368, 75025, 832040, 14930352, 267914296, 4807526976, 12586269025, 86267571272, 225851433717, 1548008755920, 27777890035288, 498454011879264, 2111485077978050, 8944394323791464, 160500643816367088, 2880067194370816120, 4660046610375530309, 51680708854858323072

Offset: 1

Asher Auel, May 20 2001

nonn

a(n) <= Fibonacci(6n) since 4 | Fibonacci(6n). Using other residue classes it can be shown that a(n) << 1.134^n. How far can this method be taken? - Charles R Greathouse IV, Dec 13 2011

			144 and 2584 are Fibonacci numbers (A000045) and are not squarefree: 144 = 2^4*3^2, 2584 = 2^3*17*19.

Amiram Eldar, Table of n, a(n) for n = 1..328 (terms 1..87 from Harry J. Smith)

Cf. A000045, A061305.

Select[Fibonacci[Range[100]],!SquareFreeQ[#]&] (* Harvey P. Dale, May 01 2018 *)

{ n=0; h=g=1; for (i=0, 375, f=g + h; h=g; g=f; if (!issquarefree(f), write("b061899.txt", n++, " ", f)) ) } \\ Harry J. Smith, Jul 28 2009

a(n) = A000045(A037917(n)). - R. J. Mathar, Jan 13 2014

A065069 Numbers n such that Fibonacci(n) is not squarefree, but for all proper divisors k of n, Fibonacci(k) is squarefree.

Original entry on oeis.org

6, 25, 56, 91, 110, 153, 406, 703, 752, 820, 915, 979, 1431, 1892, 2147, 2701, 2943, 3029, 3422, 4378, 4556, 4753, 4970, 5513, 6394, 7868, 8841, 9453, 10712, 12403, 13508, 13546, 15051, 16256, 17030, 17267, 18023, 18721, 19503, 20827, 21206

Offset: 1

Len Smiley, Nov 07 2001

nonn

These are first primitive indices m for which Fib(m) is squareful. Note that Fib(km) is divisible by Fib(m).
This sequence is closely related to A001602(n), which gives the index of the smallest Fibonacci number divisible by prime(n). It can be shown that the index of the first Fibonacci number divisible by prime(n)^2 is A001602(n)*prime(n). This sequence is the collection of numbers A001602(n)*prime(n) with multiples removed. For example, A001602(2)*prime(2) = 12, but all multiples of 12 will generated by 6, the first number in this sequence. The Mathematica code assumes that Fibonacci numbers do not have any square primitive prime factors -- an assumption whose truth is an open question. - T. D. Noe, Jul 24 2003
These are the primitive elements of A037917. - Charles R Greathouse IV, Feb 02 2014
Terms after a(12) are conjectures until the factorizations of F(1271), F(1273), etc. are completed. - Charles R Greathouse IV, Feb 02 2014
Three more factorizations are needed to get the next term: F(1423), F(1427), and F(1429). If these are each squarefree, a(13) = 1431. - Charles R Greathouse IV, Dec 09 2022

			a(1) = 6 because 2^2 divides Fibonacci(6) but no smaller Fibonacci number.

Hisanori Mishima, Factorizations of Fibonacci numbers: n=1..100, n=101..200, n=201..300, n=301..400, n=401..480.
Blair Kelly, Fibonacci and Lucas Factorizations

Cf. A037917 (all indices <= 240 for which Fib(m) is squareful).

Cf. A065106, A001602, A013929 (not squarefree).

<< NumberTheory`NumberTheoryFunctions`; a = {}; l = 0; Do[m = n; If[k = 1; While[k < l + 1 && !IntegerQ[ n/ a[[k]]], k++ ]; k > l, If[ !SquareFreeQ[ Fibonacci[n]], a = Append[a, n]; l++; Print[n]]], {n, 1, 480} ]
nLimit=50000; i=3; pMax=1; iMax=1; While[p=Transpose[FactorInteger[Fibonacci[i]]][[1, -1]]; i*ppMax, pMax=p; iMax=i]; i++ ]; nMax=PrimePi[pMax]; fs={}; Do[p=Prime[n]; k=1; found=False; While[found=(Mod[Fibonacci[k], p]==0); !found&&k*p0, j=i+1; While[j<=Length[fs], If[Mod[fs[[j]], n]==0, fs[[j]]=0]; j++ ]]; i++ ]; Select[fs, #>0&&#

PARI
```
is_A065069(n)=!fordiv(n,k,k>1 && k1 \\
```

One more term from Robert G. Wilson v, Nov 08 2001

More terms from T. D. Noe, Jul 24 2003

A065106 Smallest Fibonacci index to produce a factor p^2 (for primes p).

Original entry on oeis.org

6, 12, 25, 56, 91, 110, 153, 342, 406, 552, 703, 752, 820, 915, 930, 979

Offset: 1

Len Smiley, Nov 21 2001

nonn

Following Lucas, these might be called the prime-squared ranks of apparition.

Assuming that there are no square primitive factors in the Fibonacci sequence (an open question), then this sequence continues 1431, 1892, 2147, 2701, 2943, 3029, 3422, 3852, 4378, 4556, 4753, 4970, 5050, 5513, 6162, 6394, 6972, 7550, 7868, 8841, 8862, 9453. This is obtained by sorting the sequence prime(n)*A001602(n). - T. D. Noe, Apr 15 2004

			342 is here but not in A065069 because Fib(342) is the first Fib divisible by 19^2, but 342 is divisible by 6 and so is not a primitive index.

Cf. A065069, A037917, A065107.

Cf. A001602 (smallest m such that prime(n) divides Fibonacci(m)).

A074700 a(n) = tau(F(2^n)) where tau(x) is the number of divisors of x (A000005(x)) and F(k) the k-th Fibonacci number (A000045(k)).

Original entry on oeis.org

1, 2, 4, 8, 16, 64, 256, 1024, 8192, 131072, 1048576

Offset: 1

Benoit Cloitre, Sep 03 2002

Is there any pattern in this sequence? It seems also that tau(F(m^n)) is a power of 2 for any m, any n > 0.
F(2^n) = L(2)L(4)L(8)...L(2^(n-1)) where L(x) is the Lucas numbers. This greatly reduces the difficulty of factoring these numbers. To find a(9) one needs the factorization of F(512); this was done long ago: F(2^9) = 3 * 7 * 47 * 127 * 1087 * 2207 * 4481 * 34303 * 119809 * 73327699969 * 186812208641 * 4698167634523379875583 * 125960894984050328038716298487435392001; hence a(9) = 2^13 = 8192. Since L(512), L(1024) are completed factored the next few terms are also known. L(2048) has 1 known factor and a C411, thus the next term is at least 2^23. - Sean A. Irvine, Jun 02 2005
If no member of A037917 is a power of 2, then a(n) is a power of 2 for all n. - Charles R Greathouse IV, Apr 09 2012

Blair Kelly, Lucas number factorizations
Index to divisibility sequences

Cf. A000005, A000045, A063375, A074715.

[NumberOfDivisors(Fibonacci(2^n)): n in [1..11]]; // Vincenzo Librandi, Jun 18 2017

Table[DivisorSigma[0, Fibonacci[2^n]], {n, 11}] (* Vincenzo Librandi, Jun 18 2017 *)

a(n)=numdiv(fibonacci(2^n)) \\ Charles R Greathouse IV, Apr 09 2012

More terms from Sean A. Irvine, Jun 02 2005

A175130 Indices of Fibonacci numbers that are not cubefree.

Original entry on oeis.org

6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 125, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 246, 250, 252, 258, 264, 270, 276, 282, 288, 294, 300, 306, 312, 318, 324

Offset: 1

R. J. Mathar, Feb 16 2010

nonn

Supersequence of A037917.
Conjecture: all terms are multiples of 6 or 125. - Harvey P. Dale, Apr 28 2020
The conjecture is false. The counterexamples are 392, 784, 1183, 1210, .... . - Amiram Eldar, Oct 16 2023

			Fibonacci(125) = 5^3 * 3001 * 158414167964045700001 = A000045(125) is not cubefree, which adds 125 to the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..248
Blair Kelly, Fibonacci factorizations.

Cf. A000045, A010056, A037917, A046099, A134492, A212793.

import Data.List (findIndices)
a175130 n = a175130_list !! (n-1)
a175130_list = map (+ 1) $ findIndices ((== 0) . a212793) $ tail a000045_list
-- Reinhard Zumkeller, May 27 2012

Select[Range[350],Max[FactorInteger[Fibonacci[#]][[All,2]]]>2&] (* Harvey P. Dale, Apr 28 2020 *)

is(n)=n>5 && vecmax(factor(fibonacci(n))[,2])>2 \\ Charles R Greathouse IV, Nov 07 2014

A000045 INTERSECT A046099.

A010056(a(n)) * (1 - A212793(a(n))) = 1. - Reinhard Zumkeller, May 27 2012

A233514 Numbers n such that all numbers of the form Fib(n)/d + d are prime (or nonexistent), where Fib(n) is the n-th Fibonacci number and d is a nontrivial divisor of Fib(n).

Original entry on oeis.org

3, 4, 5, 7, 9, 11, 13, 15, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, 9311, 9677

Offset: 1

T. D. Noe, Jan 31 2014

Damir et al. conjecture that this sequence is finite.

Mohamed Taoufiq Damir, Bernadette Faye, Florian Luca, and Amadou Tall, Fibonacci numbers with prime sums of complementary divisors, Integers 14 (2014), A5.

Not a subsequence of A037917.

Cf. A001605, A080715, A233515.

f2[n_] := Module[{d = Rest[Most[Divisors[n]]]}, n/d + d]; Select[Range[3, 200], And @@ PrimeQ[f2[Fibonacci[#]]] &]

is(n)=my(F=fibonacci(n)); if(n%6==0 || n%25==0 || n%56==0 || n%91==0 || n%110==0 || n%153==0 || !issquarefree(F), return(0)); fordiv(F,d,if(d>1 && dCharles R Greathouse IV, Feb 04 2014

a(18)-a(24) from Charles R Greathouse IV, Feb 04 2014

Terms 2971 to 9677 from Don Reble. - N. J. A. Sloane, Nov 04 2022

A280297 Numbers n such that F(n) and F(n-1) are not squarefree where F(n) = A000045(n).

Original entry on oeis.org

25, 91, 126, 175, 225, 276, 325, 426, 456, 475, 576, 625, 637, 703, 726, 775, 820, 876, 925, 979, 1001, 1002, 1026, 1075, 1176, 1183, 1225, 1275, 1326, 1375, 1431, 1476, 1525, 1548, 1625, 1626, 1675, 1729, 1776, 1825, 1926, 1975, 2076, 2094, 2125, 2148, 2226

Offset: 1

Altug Alkan, Dec 31 2016

nonn

Utilized the factorization of Fibonacci numbers in the FactorDB.com database. - Robert Price, Feb 27 2017

			25 is a term because Fibonacci(25) = 5^2 * 3001 and Fibonacci(24) = 2^5 * 3^2 * 7 * 23.

Robert Price, Table of n, a(n) for n = 1..108
FactorDB, Factorizations of F(1000) ... F(1019).

Cf. A000045, A013929, A037917.

is(n) = !issquarefree(fibonacci(n)) && !issquarefree(fibonacci(n-1));

a(8)-a(108) from Robert Price, Feb 27 2017

cp's OEIS Frontend

A037918 Numbers n such that the Fibonacci number F(n) is squarefree.

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A061899 Fibonacci numbers that are not squarefree.

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A065069 Numbers n such that Fibonacci(n) is not squarefree, but for all proper divisors k of n, Fibonacci(k) is squarefree.

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A065106 Smallest Fibonacci index to produce a factor p^2 (for primes p).

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A074700 a(n) = tau(F(2^n)) where tau(x) is the number of divisors of x (A000005(x)) and F(k) the k-th Fibonacci number (A000045(k)).

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A175130 Indices of Fibonacci numbers that are not cubefree.

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A233514 Numbers n such that all numbers of the form Fib(n)/d + d are prime (or nonexistent), where Fib(n) is the n-th Fibonacci number and d is a nontrivial divisor of Fib(n).

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