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

%I A060320 #42 Feb 03 2025 14:10:50
%S A060320 1,3,8,15,20,30,40,70,60,80,90,140,176,120,168,180,324,252,240,378,
%T A060320 450,432,552,360,420,690,504,880,630,600,756,720,900,792,840,1296,
%U A060320 1050,1350,1140,1080,1200,1824,1260,1512,1320,1560,1680
%N A060320 Index of smallest Fibonacci number with exactly n distinct prime factors.
%C A060320 From _Jon E. Schoenfield_, Dec 28 2016: (Start)
%C A060320  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)
%C A060320 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
%H A060320 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fib.html">Fibonacci numbers with tables of F(0)-F(500)</a>
%H A060320 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha108.htm">Fibonacci numbers (n = 1 to 100</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha109.htm">n = 101 to 200</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha110.htm">n = 201 to 300</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha111.htm">n = 301 to 400</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha112.htm">n = 401 to 480)</a>.
%F A060320 a(n) = min (k : A022307(k) = n).
%e A060320 n=9: F(80) = 23416728348467685 = 3 * 5 * 7 * 11 * 41 * 47 * 1601 * 2161 * 3041.
%e A060320 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.
%t A060320 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 *)
%t A060320 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 *)
%o A060320 (PARI) my(o=[],s); print1(1); for(n=1,20, s=0; until( o[s]==n, #o<s++ && o=concat(o,omega(fibonacci(s))) ); print1(","s))
%Y A060320 Cf. A001605, A005478, A022307, A051694, A060319.
%Y A060320 Row n=1 of A303217.
%K A060320 nonn,more
%O A060320 0,2
%A A060320 _Labos Elemer_, Mar 28 2001
%E A060320 Corrected by _Shyam Sunder Gupta_, Jul 20 2002
%E A060320 Edited by _M. F. Hasler_, Nov 01 2012
%E A060320 a(35)-a(40), a(42), a(44) computed based on Kelly's data in A022307 by _Jon E. Schoenfield_, Dec 28 2016
%E A060320 a(41), a(43), a(45)-a(46) from _Max Alekseyev_, Feb 03 2025