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

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

Author

Labos Elemer, Mar 28 2001

Keywords

• nonn
• more

Comments

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

Examples

			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.
		

Links

• 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).

Crossrefs

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

Row n=1 of A303217.

Programs

Mathematica

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 *)

PARI

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

Formula

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

Extensions

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