cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Views

Author

Asher Auel, May 20 2001

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A000045, A061305.

Programs

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

Formula

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

A074691 Squarefree Fibonacci numbers with odd number of prime factors.

Original entry on oeis.org

2, 3, 5, 13, 89, 233, 610, 987, 1597, 10946, 28657, 514229, 3524578, 9227465, 24157817, 39088169, 63245986, 433494437, 701408733, 1134903170, 1836311903, 2971215073, 7778742049, 20365011074, 365435296162, 591286729879
Offset: 1

Views

Author

Felice Russo, Sep 03 2002

Keywords

Comments

Agrees for a long time with sequence of Fibonacci numbers whose number of divisors is a Fibonacci number.

Examples

			610 belongs to the sequence because it has 3 prime factors (2, 5, 61); it also has 8 divisors (1, 2, 5, 10, 61, 122, 305, 610).

Links

Crossrefs

Subsequence of A061305 (squarefree Fibonacci numbers).
Cf. A000045.
Cf. A022307, A075735.

Programs

  • Maple
    with(combinat, fibonacci): m2_fib := proc(n); if (numtheory[mobius](fibonacci(n))=-1) then RETURN(fibonacci(n)); fi; end: seq(m2_fib(i), i=1..100);
  • Mathematica
    Select[Fibonacci[Range[80]], MoebiusMu[#] == -1 &] (* Harvey P. Dale, Aug 23 2011 *)

Formula

Fibonacci(n) such that mu(Fibonacci(n)) = -1, where mu(n) is the Moebius mu function (A008683).

Extensions

More terms from Jani Melik, Oct 07 2002

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

Views

Author

Jani Melik, Oct 07 2002

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    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);

A075731 Fibonacci numbers F(k) for k squarefree (A005117).

Original entry on oeis.org

1, 1, 2, 5, 8, 13, 55, 89, 233, 377, 610, 1597, 4181, 10946, 17711, 28657, 121393, 514229, 832040, 1346269, 3524578, 5702887, 9227465, 24157817, 39088169, 63245986, 165580141, 267914296, 433494437, 1836311903, 2971215073
Offset: 1

Views

Author

Jani Melik, Oct 07 2002

Keywords

Examples

			10 is squarefree and fibonacci(10)=55, 11 is squarefree and fibonacci(11)=89.

Crossrefs

Cf. A000045, A005117, A061305.

Programs

  • Maple
    with(combinat, fibonacci): fib_dsk := proc(n); if (numtheory[issqrfree](n)='true') then RETURN(fibonacci(n)); fi; end: seq(fib_dsk(i), i=1..100);

A075738 Squarefree Fibonacci numbers whose indices are also squarefree.

Original entry on oeis.org

1, 1, 2, 5, 13, 55, 89, 233, 377, 610, 1597, 4181, 10946, 17711, 28657, 121393, 514229, 1346269, 3524578, 5702887, 9227465, 24157817, 39088169, 63245986, 165580141, 433494437, 1836311903, 2971215073, 20365011074, 53316291173
Offset: 1

Views

Author

Jani Melik, Oct 07 2002

Keywords

Examples

			Fib(10)=55 is there because both 10 and 55 are squarefree.

Crossrefs

Cf. A000045, A005117, A061305, A075739.

Programs

  • Maple
    with(combinat, fibonacci): fib_supM_dsk := proc(n); if (numtheory[issqrfree](n)='true') then if (numtheory[issqrfree](fibonacci(n))='true') then RETURN(fibonacci(n)); fi; fi; end: seq(fib_supM_dsk(i), i=1..100);
  • Mathematica
    Select[With[{nn=70},Thread[{Range[nn],Fibonacci[Range[nn]]}]], AllTrue[#, SquareFreeQ]&][[All,2]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 17 2018 *)

Extensions

Edited by Don Reble, Nov 05 2005

A136341 Fibonacci primes or semiprimes F(k) such that F(k+1) is again prime or semiprime.

Original entry on oeis.org

2, 3, 13, 21, 34, 55, 233, 17711
Offset: 1

Views

Author

Cino Hilliard, Mar 28 2008

Keywords

Comments

By definition, the smaller number in a pair of two consecutive Fibonacci numbers in A061305. a(9), if it exists, is >= A000045(230), so it has at least 48 digits. [R. J. Mathar, Feb 06 2010]
A search for consecutive numbers in the union of A072381 and A001605 shows that a(9) must be larger than A000045(990), a number with 207 digits, if it exists. [R. J. Mathar, Jun 02 2010]

Examples

			(55,89) is an almost twin Fibonacci prime pair because 55=5*11 is a 2-almost prime and 89 is prime.

Crossrefs

Cf. A001358.
Cf. A053409, A005478. [R. J. Mathar, Jun 02 2010]
Cf. A001605, A072381, A278637.

Programs

  • Mathematica
    Fibonacci[#]&/@(SequencePosition[Table[If[PrimeOmega[f]<=2,1,0],{f,Fibonacci[ Range[150]]}],{1,1}][[All,1]]) (* Harvey P. Dale, Mar 29 2022 *)
  • PARI
    ATfib(n) = for(x=3,n,f1=fibonacci(x);f2=fibonacci(x+1);if(bigomega (f1)<=2&&bigomega(f2)<=2, print1(f1",")))
  • PARI
    for( k=3,10^5, bigomega( fibonacci( k++ ))>2 & next; bigomega( fibonacci( k-1 ))>2 & next; print1(fibonacci(k--)",")) \\ M. F. Hasler, May 01 2008

Formula

Let F(n) = n-th Fibonacci number and define a 2-almost prime number to be a number with only 2 prime divisors with multiplicity.

Extensions

Edited by M. F. Hasler, May 01 2008

A238497 Cubefree Fibonacci numbers.

Original entry on oeis.org

1, 1, 2, 3, 5, 13, 21, 34, 55, 89, 233, 377, 610, 987, 1597, 4181, 6765, 10946, 17711, 28657, 75025, 121393, 196418, 317811, 514229, 1346269, 2178309, 3524578, 5702887, 9227465, 24157817, 39088169, 63245986, 102334155, 165580141, 433494437
Offset: 1

Views

Author

Carmine Suriano, Feb 27 2014

Keywords

Examples

			a(21) = 75025 since 75025 = fib(25) = 5^2 * 3001 has no cubic factor.

Links

Crossrefs

Cf. A000045, A061305.
Intersection of A004709 and A000045.

Programs

  • Haskell
    a238497 n = a238497_list !! (n-1)
a238497_list = filter ((== 1) . a212793) $ tail a000045_list
-- Reinhard Zumkeller, Feb 28 2014
  • Mathematica
    Select[Fibonacci@ Range@ 120, Max[Last /@ FactorInteger@ #] < 3 &] (* Giovanni Resta, Feb 27 2014 *)

Formula

A010056(a(n)) * A212793(a(n)) = 1. - Reinhard Zumkeller, Feb 28 2014

Extensions

b-file corrected from Giovanni Resta, Feb 28 2014

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

Views

Author

Vincenzo Librandi, Nov 12 2014

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Magma
    [SquareFree(Fibonacci(n^2)): n in [1..20]];
  • Mathematica
    Table[Times@@Power@@@({#[[1]], Mod[#[[2]], 2]}&/@FactorInteger[Fibonacci[n^2]]), {n, 20}]
  • PARI
    for(n=1, 60, print1(core(fibonacci(n^2)), ", "))

Formula

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

A301561 Sphenic Fibonacci numbers.

Original entry on oeis.org

610, 987, 10946, 3524578, 9227465, 24157817, 39088169, 63245986, 1836311903, 7778742049, 20365011074, 591286729879, 4052739537881, 17167680177565, 44945570212853, 61305790721611591, 420196140727489673, 1500520536206896083277, 6356306993006846248183
Offset: 1

Views

Author

Waldemar Puszkarz, Mar 23 2018

Keywords

Comments

Intersection of A000045 and A007304. There are 28 sphenic numbers among the first 200 positive Fibonacci numbers.

Examples

			610 is a term since it is a Fibonacci number that is a product of 3 distinct primes, 610=2*5*61, which makes it a sphenic number.

Crossrefs

Cf. A000045, A007304, A061305 (squarefree Fibonaccis), A137563 (Fibonaccis with 3 distinct primes).

Programs

  • Mathematica
    Select[Fibonacci@Range[120], SquareFreeQ[#]&&PrimeNu[#]==3&]
  • PARI
    for(n=1, 120, fn=fibonacci(n); issquarefree(fn)&&omega(fn)==3&&print1(fn ","))

Formula

A000045 INTERSECT A007304.
Showing 1-9 of 9 results.

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