A037918 -id:A037918 - 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

A037917 Numbers n such that the Fibonacci number F(n) is divisible by a square.

Original entry on oeis.org

6, 12, 18, 24, 25, 30, 36, 42, 48, 50, 54, 56, 60, 66, 72, 75, 78, 84, 90, 91, 96, 100, 102, 108, 110, 112, 114, 120, 125, 126, 132, 138, 144, 150, 153, 156, 162, 168, 174, 175, 180, 182, 186, 192, 198, 200, 204, 210, 216, 220, 222, 224, 225, 228, 234, 240

Offset: 1

Marc LeBrun

nonn

Is a(n) asymptotic to C*n with 4 < C < 4.5 ? - Benoit Cloitre, Sep 04 2002
Numbers are a superset of the multiples of 6 (A008588), because 8 divides Fibonacci(6m) = A134492(m). Sequence apparently also contains the multiples of 25. Are all a(n) composite? Members not divisible by 6 or 25 are 56, 91, 110, 112, 153, 182, 220, 224, 273, 280, ... - Ralf Stephan, Jan 26 2014
These numbers are the positive multiples of A065069. - Charles R Greathouse IV, Feb 02 2014
To address Cloitre's question, if such C exists it must be less than 4.3 using the known terms of A065069. - Charles R Greathouse IV, Feb 04 2014

Amiram Eldar, Table of n, a(n) for n = 1..328 (terms 1..235 from T. D. Noe, based on Kelly's table)
Blair Kelly, Fibonacci Factorizations
Eric Weisstein's World of Mathematics, Fibonacci Number.

Cf. A000045, A037918, A001177, A030426.

Select[Range[100], MoebiusMu[Fibonacci[#]] == 0 &] (* Alonso del Arte, Jan 26 2014 *)

is(n)=!issquarefree(fibonacci(n)) \\ Charles R Greathouse IV, Feb 02 2014

More terms from Eric W. Weisstein

A108120 Floor[n*1/Sin[1]], or Beatty sequence for 1/sin(1).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 83, 84, 85

Offset: 1

Zak Seidov, Jun 04 2005

nonn

Complement of A108587; not the same as A108586: a(37)=43 <> A108586(37)=44. - Reinhard Zumkeller, Jun 11 2005

Cf. A023800, A031943, A037918, A039215, A043091, A047253.

Cf. A108593.

a[n_]:=Floor[n*1/Sin[1]];Table[a[n], {n, 90}]

a(n) = floor(n*1/sin(1))

A107037 Indices of squarefree Jacobsthal numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88

Offset: 1

Paul Barry, May 09 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..899

Cf. A001045, A005117, A037918.

Rest[Flatten[Position[CoefficientList[Series[1/(1-x-2x^2),{x,0,100}],x],?SquareFreeQ]]] (* _Harvey P. Dale, May 27 2016 *)

Offset corrected by Amiram Eldar, Feb 25 2024

A107038 First differences of indices of squarefree Fibonacci numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 0

Paul Barry, May 09 2005

nonn

First differences of A037918.

Amiram Eldar, Table of n, a(n) for n = 0..1078 (terms 0..763 from Muniru A Asiru)

Cf. A000045, A037918.

P1:=List(List(List([1..180], n->Fibonacci(n)),Factors),Collected);;
P2:=Positions(List(List([1..Length(P1)],i->List([1..Length(P1[i])],j->P1[i][j][2])),Set),[1]);; a:=List([1..Length(P2)-1],j->P2[j+1]-P2[j]); # Muniru A Asiru, Jul 06 2018

with(numtheory): with(combinat): a:=proc(n) if mobius(fibonacci(n))<>0 then n else fi end:A:=[seq(a(n),n=1..180)]:seq(A[j]-A[j-1],j=2..nops(A)); # Emeric Deutsch, May 30 2005

Range[200] // Select[#, SquareFreeQ[Fibonacci[#]]&]& // Differences (* Jean-François Alcover, Aug 29 2024 *)

lista(nn) = {my(v = select(x->issquarefree(x), vector(nn, k, fibonacci(k)), 1)); vector(#v-1, k, v[k+1] - v[k]);} \\ Michel Marcus, Jul 09 2018

More terms from Emeric Deutsch, May 30 2005

A107040 Indices of squarefree Pell numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 10, 11, 13, 17, 18, 19, 22, 23, 25, 26, 27, 29, 31, 33, 34, 37, 38, 39, 41, 43, 46, 47, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 78, 79, 81, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 99, 101, 102, 103, 106, 107, 109, 110

Offset: 1

Paul Barry, May 09 2005

nonn

The associated Pell numbers are 1, 2, 5, 29, 70, 985, 2378, 5741, 33461, 1136689, 2744210, 6625109,... - R. J. Mathar, Jan 04 2017

Amiram Eldar, Table of n, a(n) for n = 1..374

Cf. A000129, A005117, A037918, A107037.

{i: A000129(i) in A005117}. - R. J. Mathar, Jan 04 2017

More terms from Amiram Eldar, Feb 25 2024

A108611 Excess of Beatty-function of 1/sin(1) over n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16

Offset: 0

Zak Seidov, Jun 13 2005

nonn

Cf. A023800, A031943, A037918, A039215, A043091, A047253, A108120.

a(n) = A108120[n] - n.

A258574 Numbers n such that Fibonacci(n)+Lucas(n) is squarefree.

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 25, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 42, 43, 45, 46, 48, 51, 52, 54, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 81, 82, 84, 85, 87, 88, 91, 93, 94, 96, 97, 100

Offset: 1

Vincenzo Librandi, Jun 01 2015

nonn

It appears that the sequence consists of the numbers congruent to 0 or 1 mod 3 (A032766) except for 24, 49, 55, 90, 99, 109, 111, ... What are these exceptions?
Also numbers n such that 2*Fibonacci(n+1) is squarefree because Lucas(n) = Fibonacci(n-1)+Fibonacci(n+1). - Michel Lagneau, Jun 04 2015
Numbers n such that Fibonacci(n+1) is odd and squarefree. - Chai Wah Wu, Jun 04 2015
Is it a theorem that this is a subsequence of A032766? - N. J. A. Sloane, Jun 04 2015
This sequence is a subsequence of A032766.  Proof: since Fibonacci(0) = 0 and Fibonacci(1) = 1, Fibonacci(n) mod 2 has the pattern: 0, 1, 1, 0, 1, 1, 0, ..., i.e. if n mod 3 = 0, then Fibonacci(n) is even, and n-1 is not a member of this sequence.  In other words, members of this sequence must be congruent to 0 or 1 mod 3. - Chai Wah Wu, Jun 04 2015

Chai Wah Wu, Table of n, a(n) for n = 1..611 (based on A037918)

Cf. A000032, A000045, A005117, A032766, A037918, A118658.

[n: n in [0..200] | IsSquarefree(Fibonacci(n)+Lucas(n))];

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

is(n)=n%3<2 && issquarefree(fibonacci(n+1)) \\ Charles R Greathouse IV, Jun 04 2015

from sympy import factorint
A258574_list = []
a, b = 0, 2
for n in range(10**2):
    if max(factorint(b).values()) <= 1:
        A258574_list.append(n)
    a, b = b, a + b # Chai Wah Wu, Jun 04 2015

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

Edited by N. J. A. Sloane, Jun 04 2015

A108612 Beatty-2 (or nested Beatty) sequence for 1/sin(1).

Original entry on oeis.org

1, 4, 9, 16, 25, 42, 56, 72, 90, 110, 143, 168, 195, 224, 255, 304, 340, 378, 418, 460, 504, 572, 621, 672, 725, 780, 864, 924, 986, 1050, 1116, 1216, 1287, 1360, 1435, 1512, 1591, 1710, 1794, 1880, 1968, 2058, 2193, 2288, 2385, 2484, 2585, 2736, 2842, 2950

Offset: 1

Zak Seidov, Jun 13 2005

nonn

Cf. A023800, A031943, A037918, A039215, A043091, A047253, A108120, A108611, A108613.

a(n) = floor(n*floor(n/sin(1))).

A108613 Excess of Beatty-2 function of 1/sin(1) over n^2.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 7, 8, 9, 10, 22, 24, 26, 28, 30, 48, 51, 54, 57, 60, 63, 88, 92, 96, 100, 104, 135, 140, 145, 150, 155, 192, 198, 204, 210, 216, 222, 266, 273, 280, 287, 294, 344, 352, 360, 368, 376, 432, 441, 450, 459, 468, 477, 540, 550, 560, 570, 580, 649, 660

Offset: 0

Zak Seidov, Jun 13 2005

nonn

Cf. A108612 Beatty-2 (or nested Beatty) function of 1/sin(1).

Cf. A023800, A031943, A037918, A039215, A043091, A047253, A108120, A108611, A108612.

a(n) = A108612[n] - n^2 = floor(n*floor(n/sin(1))) - n^2.

cp's OEIS Frontend

A061305 Squarefree Fibonacci numbers.

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A037917 Numbers n such that the Fibonacci number F(n) is divisible by a square.

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A108120 Floor[n*1/Sin[1]], or Beatty sequence for 1/sin(1).

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A107037 Indices of squarefree Jacobsthal numbers.

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A107038 First differences of indices of squarefree Fibonacci numbers.

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A107040 Indices of squarefree Pell numbers.

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A108611 Excess of Beatty-function of 1/sin(1) over n.

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A258574 Numbers n such that Fibonacci(n)+Lucas(n) is squarefree.

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Magma