A001605 -id:A001605 - cp's OEIS frontend

A107036 Indices of prime Jacobsthal numbers.

3, 4, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399

Offset: 1

Paul Barry, May 09 2005

All terms are prime except a(1) = 4. All prime terms are listed in A000978. - Alexander Adamchuk, Oct 03 2006

Cf. A000978, A001045, A001605.

is(n)=ispseudoprime((2^n - (-1)^n) / 3) \\ Charles R Greathouse IV, Apr 28 2015

a(n) = A000978(n-1) for n >= 3. - Alexander Adamchuk, Oct 03 2006

More terms from Alexander Adamchuk, Oct 03 2006

a(41)-a(42) from Bill McEachen, Aug 28 2024

A051694 Smallest Fibonacci number that is divisible by n-th prime.

Original entry on oeis.org

2, 3, 5, 21, 55, 13, 34, 2584, 46368, 377, 832040, 4181, 6765, 701408733, 987, 196418, 591286729879, 610, 72723460248141, 190392490709135, 24157817, 8944394323791464, 160500643816367088, 89, 7778742049, 12586269025

Offset: 1

N. J. A. Sloane

It is conjectured that a(n) is not divisible by prime(n)^2. See Remark on p. 528 of Wall and Conjectures in CNRS links. - Michel Marcus, Feb 24 2016

			55 is first Fibonacci number that is divisible by 11, the 5th prime, so a(5) = 55.

Alois P. Heinz, Table of n, a(n) for n = 1..650 (first 100 terms from Zak Seidov)
Shalom Eliahou, Mystères Arithmétiques de la Suite de Fibonacci, (in French), Images des Mathématiques, CNRS, 2014.
Ron Knott, Fibonacci numbers with tables of F(0)-F(500)
D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, 67 (1960), 525-532.

Cf. A000045, A001602, A001605, A005478.

F:= proc(n) option remember; `if`(n<2, n, F(n-1)+F(n-2)) end:
a:= proc(n) option remember; local p, k; p:=ithprime(n);
      for k while irem(F(k), p)>0 do od; F(k)
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 28 2015

f[n_] := Block[{fib = Fibonacci /@ Range[n^2]}, Reap@ For[k = 1, k <= n, k++, Sow@ SelectFirst[fib, Mod[#, Prime@ k] == 0 &]] // Flatten //
Rest]; f@ 26 (* Michael De Vlieger, Mar 28 2015, Version 10 *)

a(n)=if(n==3,5,my(p=prime(n));fordiv(p^2-1,d,if(fibonacci(d)%p==0, return(fibonacci(d))))) \\ Charles R Greathouse IV, Jul 17 2012

a(n) = A000045(A001602(n)). - Max Alekseyev, Dec 12 2007

log a(n) << (n log n)^2. - Charles R Greathouse IV, Jul 17 2012

More terms from Jud McCranie

More terms from James Sellers, Dec 08 1999

A060320 Index of smallest Fibonacci number with exactly n distinct prime factors.

Original entry on oeis.org

1, 3, 8, 15, 20, 30, 40, 70, 60, 80, 90, 140, 176, 120, 168, 180, 324, 252, 240, 378, 450, 432, 552, 360, 420, 690, 504, 880, 630, 600, 756, 720, 900, 792, 840, 1296, 1050, 1350, 1140, 1080, 1200, 1824, 1260, 1512, 1320, 1560, 1680

Offset: 0

Labos Elemer, Mar 28 2001

From Jon E. Schoenfield, Dec 28 2016: (Start)
 Note that the presence of incompletely factored Fibonacci numbers with indices as low as 1301 does not prevent the drawing of conclusions such as "a(44) = 1320" with certainly. Using F(1301) as an example, the compact table of Fibonacci results at the Kelly site indicates that F(1301) = p*q*r*c where p=6400921, q=14225131397, r=100794731109596201, and c is a 238-digit unfactored composite number. The complete factorization of every Fibonacci number up to F(1000) is explicitly given elsewhere on the site, and those results allow quick verification that a(n) <= 900 for all n in [0..34], so 1301 cannot be a term unless F(1301) has at least 35 distinct prime factors, which would require c to have at least 32 distinct prime factors, at least one of which would have to be less than ceiling(c^(1/32)) = 26570323, but trial division of c by every prime less than 26570323 shows that c has no prime factors that small. Thus, while A022307(1301) is unknown, it is certain that 1301 is not a term in this sequence. Similarly, making use of known factors, it can be proved that F(n) cannot have 44 or more distinct prime factors for any n < 1320, so since F(1320) has exactly 44 distinct prime factors, it is established that a(44) = 1320. (End)
a(47) >= 2835, a(48..68) = (2040, 1800, 2736, 2730, 1890, 1980, 2520, 2280, 2100, 2160, 2640, 3300, 3060, 3150, 2520, 3120, 3696, 3240, 3990, 3360, 3420), a(69) >= 4400, a(75) = 4320, a(77) = 4200, a(79) = 3780. - Max Alekseyev, Feb 03 2025

			n=9: F(80) = 23416728348467685 = 3 * 5 * 7 * 11 * 41 * 47 * 1601 * 2161 * 3041.
n=25: F(690) = 2^3 * 5 * 11 * 31 * 61 * 137 * 139 * 461 * 691 * 829 * 1151 * 1381 * 4831 * 5981 * 18077 * 28657 * 186301 * 324301 * 686551 * 1485571 * 4641631 * 117169733521 * 2441738887963981 * 3490125311294161 * 25013864044961447973152814604981 is the smallest Fibonacci number with exactly 25 distinct prime factors.

Ron Knott, Fibonacci numbers with tables of F(0)-F(500)
Hisanori Mishima, Fibonacci numbers (n = 1 to 100, n = 101 to 200, n = 201 to 300, n = 301 to 400, n = 401 to 480).

Cf. A001605, A005478, A022307, A051694, A060319.

Row n=1 of A303217.

First /@ SortBy[#, Last] &@ Map[First@ # &, Values@ GroupBy[#, Last]] &@ Table[{n - Boole[n == 2], #, PrimeNu@ #} &@ Fibonacci@ n, {n, 2, 300}] (* Michael De Vlieger, Feb 18 2017, Version 10 *)
Module[{ff=Table[{n,PrimeNu[Fibonacci[n]]},{n,1400}]},Table[ SelectFirst[ ff,#[[2]]==k&],{k,0,40}]][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 28 2018 *)

my(o=[],s); print1(1); for(n=1,20, s=0; until( o[s]==n, #o

a(n) = min (k : A022307(k) = n).

Corrected by Shyam Sunder Gupta, Jul 20 2002
Edited by M. F. Hasler, Nov 01 2012
a(35)-a(40), a(42), a(44) computed based on Kelly's data in A022307 by Jon E. Schoenfield, Dec 28 2016
a(41), a(43), a(45)-a(46) from Max Alekseyev, Feb 03 2025

A060385 Largest prime factor of n-th Fibonacci number.

Original entry on oeis.org

2, 3, 5, 2, 13, 7, 17, 11, 89, 3, 233, 29, 61, 47, 1597, 19, 113, 41, 421, 199, 28657, 23, 3001, 521, 109, 281, 514229, 61, 2417, 2207, 19801, 3571, 141961, 107, 2221, 9349, 135721, 2161, 59369, 421, 433494437, 307, 109441, 28657, 2971215073, 1103

Offset: 3

Labos Elemer, Apr 03 2001

nonn

For n > 12, Fibonacci(n) is divisible by a primitive prime factor (one not dividing Fibonacci(1), ..., Fibonacci(n-1)). But all primes up to n-2 divide smaller Fibonacci numbers, see A001602, so a(n) >= n-1 for n > 12. This strengthens a more general theorem of Bravo and Luca. - Charles R Greathouse IV, Feb 01 2013

			F(82) = 2789 * 59369 * 370248451, so a(82) = 370248451.

Tyler Busby, Table of n, a(n) for n = 3..1422 (terms 3..1000 from Charles R Greathouse IV, terms 1001..1408 from Amiram Eldar)
U. Alfred, On the form of primitive factors of Fibonacci numbers, Fibonacci Quarterly 1:1 (1963), pp. 43-45.
Jhon J. Bravo and Florian Luca, On the largest prime factor of the k-Fibonacci numbers, arXiv:1210.4101 [math.NT], 2012.
D. E. Daykin and L. A. G. Dresel, Factorization of Fibonacci numbers, Fibonacci Quarterly 8:1 (1970), pp. 23-30.

Cf. A000045, A022307, A001605, A060319, A060320, A051694, A193615.

[Maximum(PrimeDivisors(Fibonacci(n))): n in [3..50]]; // Vincenzo Librandi, Dec 25 2016

Table[First[Last[FactorInteger[Fibonacci[n]]]], {n, 3, 100}] (* Vladimir Joseph Stephan Orlovsky, Feb 03 2012 *)

a(n)=my(f=factor(fibonacci(n))[,1]);f[#f] \\ Charles R Greathouse IV, Feb 01 2013

a(n) >= n - 1 for n > 12, see comments. It is not hard to show that a(n) > 1000 for n > 88. Similarly a(n) > 20641 for n > 120. - Charles R Greathouse IV, Feb 01 2013

A113188 Primes that are the difference of two Fibonacci numbers; primes in A007298.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 29, 31, 47, 53, 89, 131, 139, 199, 233, 521, 607, 953, 1453, 1597, 2207, 2351, 2579, 3571, 6763, 9349, 10891, 28513, 28649, 28657, 42187, 44771, 46279, 75017, 189653, 317777, 514229, 1981891, 2177699, 3010349, 3206767

Offset: 1

T. D. Noe, Oct 17 2005

nonn

The difference F(i)-F(j) equals the sum F(j-1)+...+F(i-2) [Corrected by Patrick Capelle, Mar 01 2008]. In general, we need gcd(i,j)=1 for F(i)-F(j) to be prime. The exceptions are handled by the following rule: if i and j are both even or both odd, then F(i)-F(j) is prime if either (1) i-j=4 and L(i-2) is a Lucas prime or (2) i-j=2 and F(i-1) is a Fibonacci prime.

			The prime 139 is here because it is F(12)-F(5).

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000045 (Fibonacci numbers), A001605 (Fibonacci(n) is prime), A001606 (Lucas(n) is prime), A113189 (number of times that Fibonacci(n)-Fibonacci(i) is prime for i=0..n-3).

lst={}; Do[p=Fibonacci[n]-Fibonacci[i]; If[PrimeQ[p], AppendTo[lst, p]], {n, 2, 40}, {i, n-1}]; Union[lst]
Select[Union[Flatten[Differences/@Subsets[Fibonacci[Range[50]],{2}]]],PrimeQ] (* Harvey P. Dale, Aug 04 2024 *)

list(lim)=my(v=List(),F=vector(A130233(lim),i,fibonacci(i)),s,t); for(i=1,#F, s=0; forstep(j=i,1,-1, s+=F[j]; if(s>lim, break); if(isprime(s), listput(v,s)))); Set(v) \\ Charles R Greathouse IV, Oct 07 2016

A119984 Numbers k such that Fibonacci(prime(k)) is prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 14, 15, 23, 32, 33, 72, 83, 84, 87, 97, 104, 105, 429, 637, 710, 1152, 1194, 1692, 2814, 3316, 3824, 3971, 5206, 8002, 10016, 12161, 13681, 18069, 33653, 36467, 48355, 48629, 49455, 73574, 82128, 99005, 123685, 135276, 146779, 210404, 233207, 239581

Offset: 1

Alexander Adamchuk, Aug 04 2006

nonn

All prime Fibonacci numbers have prime indices, except prime F(4) = 3; a(n) is such that Fibonacci(prime(a(n))) is prime. - Robert G. Wilson v, Aug 05 2006

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

Cf. A000045, A000720, A001605, A005478, A030426, A075737, A083668.

Select[ Range@3000, PrimeQ@ Fibonacci@ Prime@ # &] (* Robert G. Wilson v, Aug 05 2006 *)

a(n) = pi(A001605(n+1)). This holds for all n including n=1, since pi(4) = pi(3) = 2. - Jens Kruse Andersen, Jul 24 2014

a(21)-a(27) from Robert G. Wilson v, Aug 05 2006
More terms (from A001605) from T. D. Noe, Aug 18 2006
a(42)-a(48) (from A001605, found by Henri Lifchitz) from Jens Kruse Andersen, Jul 24 2014
a(49)-a(50) (from A001605, found by Henri Lifchitz) from Amiram Eldar, Sep 01 2019

A153862 Smallest of four consecutive Fibonacci numbers such that sum of four consecutive Fibonacci numbers is prime number.

Original entry on oeis.org

1, 1, 3, 5, 21, 55, 233, 377, 987, 317811, 5702887, 39088169, 701408733, 12586269025, 591286729879, 72723460248141, 3416454622906707, 43566776258854844738105

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

nonn

1+1+2+3=7, 1+2+3+5=11, 3+5+8+13=29, 5+8+13+21=47, 21+34+55+89=199,...

Cf. A000045, A001906, A000071, A001605

a=0;b=1;c=1;lst={};Do[d=Fibonacci[n];p=a+b+c+d;If[PrimeQ[p],AppendTo[lst,a]];a=b;b=c;c=d,{n,3,6!}];lst
Transpose[Select[Partition[Fibonacci[Range[500]],4,1],PrimeQ[ Total[ #]]&]] [[1]] (* Harvey P. Dale, Jan 11 2013 *)

A153863 Second-to-smallest of four consecutive Fibonacci numbers such that sum of four consecutive Fibonacci numbers is prime number.

Original entry on oeis.org

1, 2, 5, 8, 34, 89, 377, 610, 1597, 514229, 9227465, 63245986, 1134903170, 20365011074, 956722026041, 117669030460994, 5527939700884757, 70492524767089125814114, 44225333398004061429732838340729878012027363723832270745251370289

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

nonn

1+1+2+3=7, 1+2+3+5=11, 3+5+8+13=29, 5+8+13+21=47, 21+34+55+89=199,...

Harvey P. Dale, Table of n, a(n) for n = 1..27

Cf. A000045, A001906, A000071, A001605, A153862

a=0;b=1;c=1;lst={};Do[d=Fibonacci[n];p=a+b+c+d;If[PrimeQ[p],AppendTo[lst,b]];a=b;b=c;c=d,{n,3,6!}];lst
Select[Partition[Fibonacci[Range[200]],4,1],PrimeQ[Total[#]]&][[All,2]] (* Harvey P. Dale, Aug 28 2016 *)

One more term from Harvey P. Dale, Aug 28 2016

A135953 (Nonprime Fibonacci numbers with prime indices) that have exactly 2 prime factors.

Original entry on oeis.org

4181, 1346269, 165580141, 53316291173, 956722026041, 2504730781961, 308061521170129, 806515533049393, 14472334024676221, 1779979416004714189, 573147844013817084101, 10284720757613717413913, 26925748508234281076009

Offset: 1

Artur Jasinski, Dec 08 2007

nonn

Conjecture: All numbers in this sequence are products of two sums of two squares, e.g. 4181 = 37*113 = (1^2+6^2)*(7^2+8^2), 1346269 = 557*2417 = (14^2+19^2)*(4^2+49^2).

Amiram Eldar, Table of n, a(n) for n = 1..42

Cf. A000045, A001605, A050937, A075737, A090819, A134787, A134851, A134852, A135954, A135955, A135956, A135957.

k = {}; Do[If[ !PrimeQ[Fibonacci[Prime[n]]], c = Length[FactorInteger[Fibonacci[Prime[n]]]]; If[c == 2, AppendTo[k, Fibonacci[Prime[n]]]]], {n, 1, 50}]; k
Select[Fibonacci[Prime[Range[30]]],PrimeOmega[#]==2&] (* Harvey P. Dale, Feb 18 2012 *)

A135957 a(n) = smallest k such that Fibonacci(prime(k)) has exactly n prime factors.

Original entry on oeis.org

1, 2, 8, 12, 25, 50, 96, 73, 164

Offset: 0

Artur Jasinski, Dec 08 2007

A135958 gives prime(k). Cf. A000045, A001605, A050937, A075737, A134787, A134852.

Edited and extended by David Wasserman, Mar 26 2008

cp's OEIS Frontend

A107036 Indices of prime Jacobsthal numbers.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

Extensions

A051694 Smallest Fibonacci number that is divisible by n-th prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A060320 Index of smallest Fibonacci number with exactly n distinct prime factors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A060385 Largest prime factor of n-th Fibonacci number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A113188 Primes that are the difference of two Fibonacci numbers; primes in A007298.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A119984 Numbers k such that Fibonacci(prime(k)) is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A153862 Smallest of four consecutive Fibonacci numbers such that sum of four consecutive Fibonacci numbers is prime number.

Views

Author

Keywords

Comments

Crossrefs

Programs