A007570 -id:A007570

A153385

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

+20
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

LINKS

Table of n, a(n) for n=0..11.

Harry J. Smith, XICalc - Extra Precision Integer Calculator [broken link]

Kim Walisch, Fast C++ prime counting function implementation (primecount).

FORMULA

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

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

EXAMPLE

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

Cf. A000045, A000720, A007570, A054782.

KEYWORD

nonn,more,hard

AUTHOR

Harry J. Smith, Dec 25 2008

EXTENSIONS

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

STATUS

approved

A005371

a(n) = L(L(n)), where L(n) are Lucas numbers A000032.
(Formerly M3315)

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

REFERENCES

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).

LINKS

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

MAPLE

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

MATHEMATICA

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 *)

PROG

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

(PARI) {lucas(n) = fibonacci(n+1) + fibonacci(n-1)};

for(n=0, 15, print1(lucas(lucas(n)), ", ")) \\ G. C. Greubel, Dec 21 2017

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

CROSSREFS

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

KEYWORD

easy,nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

Offset changed Feb 28 2007

STATUS

approved

A263101

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

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

FORMULA

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

MAPLE

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);

MATHEMATICA

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 *)

CROSSREFS

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

KEYWORD

nonn,look,changed

AUTHOR

Alois P. Heinz, Oct 09 2015

STATUS

approved

A263112

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

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

FORMULA

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

MAPLE

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);

MATHEMATICA

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 *)

CROSSREFS

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

KEYWORD

nonn,look,changed

AUTHOR

Alois P. Heinz, Oct 09 2015

STATUS

approved

A058051

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

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

LINKS

Table of n, a(n) for n=0..7.

MAPLE

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A000045, A007570, A262361.

KEYWORD

easy,nonn

AUTHOR

Robert G. Wilson v, Nov 18 2000

EXTENSIONS

Offset corrected by Alois P. Heinz, Nov 11 2015

STATUS

approved

A127787

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..55.

EXAMPLE

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

MAPLE

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

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Alexander Adamchuk, May 13 2007

EXTENSIONS

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

STATUS

approved

A274996

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

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

FORMULA

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

MAPLE

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);

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Nov 11 2016

STATUS

approved

A005370

a(n) = Fibonacci(Fibonacci(n+1) + 1).
(Formerly M0891)

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

REFERENCES

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

LINKS

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

MAPLE

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

Cf. A000045, A007570.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David W. Wilson

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

STATUS

approved

A111425

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

LINKS

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

FORMULA

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

MAPLE

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

CROSSREFS

Cf. A000045, A000073, A007570, A111431.

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Nov 13 2005

STATUS

approved

A111431

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

LINKS

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

FORMULA

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

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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

CROSSREFS

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

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Nov 13 2005

STATUS

approved