A007570 -id:A007570 - cp's OEIS frontend

A153385 Number of primes <= Fibonacci(Fibonacci(n)) = pi(A007570(n)).

0, 0, 0, 0, 1, 3, 8, 51, 1329, 393790, 5670112879, 43416847208976911

Offset: 0

Harry J. Smith, Dec 25 2008

			a(7) = 51 because Fibonacci(7) = 13, Fibonacci(13) = 233 and there are 51 primes <= 233.

Harry J. Smith, XICalc - Extra Precision Integer Calculator [broken link]
Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf. A000045, A000720, A007570, A054782.

[0] cat [#PrimesUpTo(Fibonacci(Fibonacci(n))): n in [1..9]]; // Vincenzo Librandi, Aug 02 2015

PrimePi@# & /@ (Fibonacci@Fibonacci@# & /@ Range@10) (* Robert G. Wilson v, Feb 17 2009 *)

XiCalc
```
Pi(Fib(Fib(n)));
```

a(n) = pi(Fibonacci(Fibonacci(n))) = A000720(A007570(n)).

a(n) = A054782(A000045(n)). - Amiram Eldar, Sep 03 2024

a(11) calculated using Kim Walisch's primecount and added by Amiram Eldar, Sep 03 2024

A005371 a(n) = L(L(n)), where L(n) are Lucas numbers A000032.

Original entry on oeis.org

3, 1, 4, 7, 29, 199, 5778, 1149851, 6643838879, 7639424778862807, 50755107359004694554823204, 387739824812222466915538827541705412334749, 19679776435706023589554719270187913247121278789615838446937339578603

Offset: 0

N. J. A. Sloane

T. Koshy (2001), Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, New York, 511-516
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..15

Cf. A000032, A005372, A007570, A068096, A068098.

[ Lucas(Lucas(n)): n in [0..20]]; // Vincenzo Librandi, Apr 16 2011

L:= n-> (<<0|1>, <1|1>>^n. <<2,1>>)[1,1]:
a:= n-> L(L(n)):
seq(a(n), n=0..14);  # Alois P. Heinz, Jun 01 2016

l[n_]:= l[n]= l[n-1] + l[n-2]; l[0]= 2; l[1]= 1; Table[l[l[n]], {n,0,12}]
LucasL[LucasL[Range[0, 15]]] (* G. C. Greubel, Dec 21 2017 *)

{lucas(n) = fibonacci(n+1) + fibonacci(n-1)};
for(n=0,15, print1(lucas(lucas(n)), ", ")) \\ G. C. Greubel, Dec 21 2017

[lucas_number2(lucas_number2(n, 1,-1),1,-1) for n in range(15)] # G. C. Greubel Nov 14 2022

More terms from Mario Catalani (mario.catalani(AT)unito.it), Mar 14 2003

Offset changed Feb 28 2007

A263101 a(n) = F(F(n)) mod F(n), where F = Fibonacci = A000045.

Original entry on oeis.org

0, 0, 1, 2, 0, 5, 12, 5, 33, 5, 1, 0, 232, 233, 55, 5, 1596, 2563, 1, 5, 987, 10946, 28656, 0, 0, 75025, 189653, 89, 1, 6765, 1, 5, 6765, 1, 9227460, 0, 24157816, 1, 63245985, 5, 1, 267914275, 433494436, 4181, 1134896405, 1, 2971215072, 0, 7778741816, 75025

Offset: 1

Alois P. Heinz, Oct 09 2015

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Cf. A000045, A002708, A007570, A076240, A127787 (where a(n)=0), A263112, A274996.

F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
p:= (M, n, k)-> map(x-> x mod k, `if`(n=0, <<1|0>, <0|1>>,
          `if`(n::even, p(M, n/2, k)^2, p(M, n-1, k).M))):
a:= n-> p(<<0|1>, <1|1>>, F(n)$2)[1, 2]:
seq(a(n), n=1..50);

F[n_] := MatrixPower[{{0, 1}, {1, 1}}, n][[1, 2]];
p[M_, n_, k_] := Mod[#, k]& /@ If[n == 0, {{1, 0}, {0, 1}}, If[EvenQ[n], MatrixPower[p[M, n/2, k], 2], p[M, n - 1, k].M]];
a[n_] := p[{{0, 1}, {1, 1}}, F[n], F[n]][[1, 2]];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Oct 29 2024, after Alois P. Heinz *)

alist(nn)= my(f=fibonacci); [ f(f(n))%f(n) |n<-[1..nn] ]; \\ Ruud H.G. van Tol, Dec 13 2024

a(n) = A007570(n) mod A000045(n).

A263112 a(n) = F(F(n)) mod n, where F = Fibonacci = A000045.

Original entry on oeis.org

0, 1, 1, 2, 0, 3, 2, 2, 1, 5, 1, 0, 8, 13, 10, 2, 12, 15, 5, 10, 1, 1, 1, 0, 0, 25, 1, 2, 5, 15, 27, 2, 10, 33, 20, 0, 1, 1, 34, 10, 40, 21, 18, 2, 10, 1, 1, 0, 1, 25, 1, 2, 16, 21, 5, 26, 37, 1, 7, 0, 33, 27, 1, 2, 40, 21, 5, 2, 1, 15, 1, 0, 46, 1, 25, 2, 68

Offset: 1

Alois P. Heinz, Oct 09 2015

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000045, A002708, A007570, A023172 (where a(n)=0), A263101, A274996, A338736.

F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
p:= (M, n, k)-> map(x-> x mod k, `if`(n=0, <<1|0>, <0|1>>,
          `if`(n::even, p(M, n/2, k)^2, p(M, n-1, k).M))):
a:= n-> p(<<0|1>, <1|1>>, F(n), n)[1, 2]:
seq(a(n), n=1..80);

F[n_] := MatrixPower[{{0, 1}, {1, 1}}, n][[1, 2]];
p[M_, n_, k_] := Mod[#, k]& /@ If[n == 0, {{1, 0}, {0, 1}}, If[EvenQ[n], MatrixPower[p[M, n/2, k], 2], p[M, n - 1, k].M]];
a[n_] := p[{{0, 1}, {1, 1}}, F[n], n][[1, 2]];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Oct 29 2024, after Alois P. Heinz *)

a(n) = A007570(n) mod n.

A058051 a(n) = F(F(F(n))), where F is a Fibonacci number (A000045).

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 10946, 2211236406303914545699412969744873993387956988653

Offset: 0

Robert G. Wilson v, Nov 18 2000

a(8) = 1695216512..7257812353 has 2288 decimal digits and a(9) = 3525796792..4659808333 has 1191833 decimal digits. - Alois P. Heinz, Nov 11 2015

Cf. A000045, A007570, A262361.

F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
a:= n-> F(F(F(n))):
seq(a(n), n=0..7);  # Alois P. Heinz, Nov 11 2015

F[0] = 0; F[1] = 1; F[n_] := F[n] = F[n - 1] + F[n - 2]; Table[ F[ F[ F[n] ] ], {n, 0, 10} ]
Table[Nest[Fibonacci,n,3],{n,0,8}] (* Harvey P. Dale, Feb 09 2018 *)

Offset corrected by Alois P. Heinz, Nov 11 2015

A127787 Numbers n such that F(n) divides F(F(n)), where F(n) is a Fibonacci number.

Original entry on oeis.org

1, 2, 5, 12, 24, 25, 36, 48, 60, 72, 96, 108, 120, 125, 144, 168, 180, 192, 216, 240, 288, 300, 324, 336, 360, 384, 432, 480, 504, 540, 552, 576, 600, 612, 625, 648, 660, 672, 684, 720, 768, 840, 864, 900, 960, 972, 1008, 1080, 1104, 1152, 1176, 1200, 1224, 1296, 1320

Offset: 1

Alexander Adamchuk, May 13 2007

nonn

It is known that for n > 2 Fibonacci(n) divides Fibonacci(m) if and only if n divides m. Therefore if the term "2" is omitted this is identical to A023172, which see for further information. - Stefan Steinerberger, Dec 20 2007

			12 is a term because F(12) = 144 divides F(F(12)) = F(144) = 555565404224292694404015791808.

Cf. A023172. Cf. also A000045 = Fibonacci(n), A007570 = F(F(n)), where F is a Fibonacci number, A023172 = numbers n such that n divides Fibonacci(n).

Cf. A263101.

with(combinat): a:=proc(n) if type(fibonacci(fibonacci(n))/fibonacci(n), integer) then n else end if end proc: seq(a(n),n=1..40); # Emeric Deutsch, Aug 24 2007

Edited by N. J. A. Sloane, Dec 22 2007

A274996 a(n) = F(F(F(n))) mod F(F(n)), where F = Fibonacci = A000045.

Original entry on oeis.org

0, 0, 0, 1, 0, 5, 232, 987, 1, 5, 1, 0, 2211236406303914545699412969744873993387956988652, 2211236406303914545699412969744873993387956988653, 139583862445

Offset: 1

Alois P. Heinz, Nov 11 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..19

Cf. A000045, A007570, A058051, A263101, A263112, A338889.

F:= proc(n) local r, M, p; r, M, p:=
      <<1|0>, <0|1>>, <<0|1>, <1|1>>, n;
      do if irem(p, 2, 'p')=1 then r:=
        `if`(nargs=1, r.M, r.M mod args[2]) fi;
         if p=0 then break fi; M:=
        `if`(nargs=1, M.M, M.M mod args[2])
      od; r[1, 2]
    end:
a:= n-> (h-> F(h$2))(F(F(n))):
seq(a(n), n=1..15);

a(n) = A058051(n) mod A007570(n).

A005370 a(n) = Fibonacci(Fibonacci(n+1) + 1).

Original entry on oeis.org

1, 1, 2, 3, 8, 34, 377, 17711, 9227465, 225851433717, 2880067194370816120, 898923707008479989274290850145, 3577855662560905981638959513147239988861837901112, 4444705723234237498833973519982908519933430818636409166351397897095281987215864

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..18

Cf. A000045, A007570.

[Fibonacci(Fibonacci(n+1)+1): n in [0..17]]; // Vincenzo Librandi, Apr 20 2011

with(combinat, fibonacci): A005370 := n -> fibonacci(fibonacci(n+1)+1);
# second Maple program:
F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
a:= n-> F(F(n+1)+1):
seq(a(n), n=1..14);  # Alois P. Heinz, Nov 05 2015

Table[Fibonacci[Fibonacci[n+1] +1], {n, 0, 14}] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)

[fibonacci(fibonacci(n+1) +1) for n in range(15)] # G. C. Greubel, Nov 14 2022

More terms from David W. Wilson

Description corrected by Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 17 2002

A111425 a(n) = tribonacci(Fibonacci(n)).

Original entry on oeis.org

0, 0, 0, 1, 1, 4, 24, 504, 66012, 181997601, 65720971788709, 65431225571591367370292, 23523635785731871586396890786299881280, 8419860898569880503664421048610377961601349941695806840602396

Offset: 0

Jonathan Vos Post, Nov 13 2005

Alois P. Heinz, Table of n, a(n) for n = 0..18

Cf. A000045, A000073, A007570, A111431.

a:= n-> (<<0|1|0>, <0|0|1>, <1|1|1>>^((<<0|1>, <1|1>>^n)[1, 2]))[1, 3]:
seq(a(n), n=0..14);  # Alois P. Heinz, Aug 09 2018

a(n) = A000073(A000045(n)).

A111431 a(n) = Fibonacci(tribonacci(n)).

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 13, 233, 46368, 701408733, 37889062373143906, 6161314747715278029583501626149, 818706854228831001753880637535093596811413714795418360007

Offset: 0

Jonathan Vos Post, Nov 13 2005

			a(0) = Fibonacci(tribonacci(0)) = A000045(A000073(0)) = A000045(0) = 0.
a(1) = Fibonacci(tribonacci(1)) = A000045(A000073(1)) = A000045(0) = 0.
a(2) = Fibonacci(tribonacci(2)) = A000045(A000073(2)) = A000045(1) = 1.
a(3) = Fibonacci(tribonacci(3)) = A000045(A000073(3)) = A000045(1) = 1.
a(4) = Fibonacci(tribonacci(4)) = A000045(A000073(4)) = A000045(2) = 1.
a(5) = Fibonacci(tribonacci(5)) = A000045(A000073(5)) = A000045(4) = 3.
a(6) = Fibonacci(tribonacci(6)) = A000045(A000073(6)) = A000045(7) = 13.
a(7) = Fibonacci(tribonacci(7)) = A000045(A000073(7)) = A000045(13) = 233.
a(8) = A000045(A000073(8)) = A000045(24) = 46368.
a(9) = A000045(A000073(9)) = A000045(44) = 701408733.
a(10) = A000045(A000073(10)) = A000045(81) = 37889062373143906.

Alois P. Heinz, Table of n, a(n) for n = 0..16

Cf. A000045, A000073, A066178, A007570, A111425.

a:= n-> (<<0|1>, <1|1>>^((<<0|1|0>, <0|0|1>, <1|1|1>>^n)[1, 3]))[1, 2]:
seq(a(n), n=0..13);  # Alois P. Heinz, Aug 09 2018

Fibonacci/@LinearRecurrence[{1,1,1},{0,0,1},15] (* Harvey P. Dale, Jan 04 2013 *)

a(n) = A000045(A000073(n)).

cp's OEIS Frontend

A153385 Number of primes <= Fibonacci(Fibonacci(n)) = pi(A007570(n)).

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A005371 a(n) = L(L(n)), where L(n) are Lucas numbers A000032.

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A263101 a(n) = F(F(n)) mod F(n), where F = Fibonacci = A000045.

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A263112 a(n) = F(F(n)) mod n, where F = Fibonacci = A000045.

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A058051 a(n) = F(F(F(n))), where F is a Fibonacci number (A000045).

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A127787 Numbers n such that F(n) divides F(F(n)), where F(n) is a Fibonacci number.

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A274996 a(n) = F(F(F(n))) mod F(F(n)), where F = Fibonacci = A000045.

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A005370 a(n) = Fibonacci(Fibonacci(n+1) + 1).

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