A060320 -id:A060320 - cp's OEIS frontend

A022307 Number of distinct prime factors of n-th Fibonacci number.

0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 3, 3, 1, 3, 2, 4, 3, 2, 1, 4, 2, 2, 4, 4, 1, 5, 2, 4, 3, 2, 3, 5, 3, 3, 3, 6, 2, 5, 1, 5, 5, 3, 1, 6, 3, 5, 3, 4, 2, 6, 4, 6, 5, 3, 2, 8, 2, 3, 5, 6, 3, 5, 3, 5, 5, 7, 2, 8, 2, 4, 5, 5, 4, 6, 2, 9, 7, 3, 1, 9, 4, 3, 4, 9, 2, 10, 4, 6, 4, 2, 6, 9, 4, 5, 6

Offset: 0

Clark Kimberling

nonn

Although every prime divides some Fibonacci number, this is not true for the Lucas numbers. Exactly 1/3 of all primes do not divide any Lucas number. See Lagarias and Moree for more details. - Jonathan Vos Post, Dec 06 2006
First occurrence of k: 0, 3, 8, 15, 20, 30, 40, 70, 60, 80, 90, 140, 176, 120, 168, 180, 324, 252, 240, 378, ..., . - Robert G. Wilson v, Dec 10 2006 [Other than 0, this is sequence A060320. - Jon E. Schoenfield, Dec 30 2016]
Row lengths of table A060442. - Reinhard Zumkeller, Aug 30 2014
If k properly divides n then a(n) >= a(k) + 1, except for a(6) = a(3) = 1. - Robert Israel, Aug 18 2015

Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, The Fibonacci Association, 1972, pages 1-8.

Max Alekseyev, Table of n, a(n) for n = 0..1422 (terms 0..1000 and 1001..1408 from T. D. Noe and Amiram Eldar, respectively).
Blair Kelly, Fibonacci and Lucas Factorizations
J. C. Lagarias, The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math., 118. No. 2, (1985), 449-461.
J. C. Lagarias, Errata to: The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math., 162, No. 2, (1994), 393-396.
Hisanori Mishima, WIFC (World Integer Factorization Center), Fibonacci numbers (n = 1 to 100, n = 101 to 200, n = 201 to 300, n = 301 to 400, n = 401 to 480).
Pieter Moree, Counting Divisors of Lucas Numbers, Pacific J. Math, Vol. 186, No. 2, 1998, pp. 267-284.
Eric Weisstein's World of Mathematics, Fibonacci Number

Cf. A038575 (number of prime factors, counting multiplicity), A086597 (number of primitive prime factors).
Cf. A000032, A000040, A000045, A001221, A053028.
Cf. A060442, A086598 (omega(Lucas(n))).
Cf. A060320. - Jon E. Schoenfield, Dec 30 2016

a022307 n = if n == 0 then 0 else a001221 $ a000045 n
-- Reinhard Zumkeller, Aug 30 2014

[0] cat [#PrimeDivisors(Fibonacci(n)): n in [1..100]]; // Vincenzo Librandi, Jul 26 2017

Table[Length[FactorInteger[Fibonacci[n]]], {n, 150}]

a(n)=omega(fibonacci(n)) \\ Charles R Greathouse IV, Feb 03 2014

a(n) = Sum{d|n} A086597(d), Mobius transform of A086597.

a(n) = A001221(A000045(n)) = omega(F(n)). - Jonathan Vos Post, Dec 06 2006

A060384 Number of decimal digits in n-th Fibonacci number.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18

Offset: 0

Labos Elemer, Apr 03 2001

Harry J. Smith, Table of n, a(n) for n = 0..10000

Cf. A000045, A022307, A001605, A060319, A060320, A051694, A050815.

Cf. A261585.

a060384 = a055642 . a000045  -- Reinhard Zumkeller, Mar 09 2013

with(combinat): a:=n->nops(convert(fibonacci(n),base,10)): 1,seq(a(n),n=1..100); # Emeric Deutsch, May 19 2007

Table[IntegerLength@ Fibonacci@ n, {n, 0, 84}] /. 0 -> 1 (* or *)
Table[Floor[n Log10@ GoldenRatio - Log10@ 5/2] + 1, {n, 0, 84}] /. 0 -> 1 (* Michael De Vlieger, Jul 04 2016 *)

print1("1, 1, "); gold=(1+sqrt(5))/2; for(n=2,100,print1(floor((n*log(gold)-log(5)/2)/log(10))+1", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007

a(n) = #Str(fibonacci(n)); \\ Michel Marcus, Jul 04 2016

a(n) = floor(n*log(tau)/log(10)) +0 or +1 where tau is the golden ratio. - Benoit Cloitre, Oct 29 2002. [Corrected by Hans J. H. Tuenter, Jul 07 2025].
a(n) = floor(n*log_10(gold) - log_10(5)/2) + 1 for n >= 2, where gold is (1+sqrt(5))/2. - Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007
a(n) = A055642(A000045(n)). - Reinhard Zumkeller, Mar 09 2013

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007

A303217 A(n,k) is the n-th index of a Fibonacci number with exactly k distinct prime factors; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

3, 8, 4, 15, 9, 5, 20, 16, 10, 6, 30, 24, 18, 12, 7, 40, 36, 27, 21, 14, 11, 70, 48, 42, 28, 33, 19, 13, 60, 81, 54, 44, 32, 35, 22, 17, 80, 72, 104, 56, 45, 52, 37, 25, 23, 90, 84, 110, 105, 64, 50, 55, 38, 26, 29, 140, 126, 88, 112, 136, 78, 57, 74, 39, 31, 43

Offset: 1

Alois P. Heinz, Apr 19 2018

			Square array A(n,k) begins:
   3,  8, 15, 20, 30,  40,  70,  60,  80,  90, ...
   4,  9, 16, 24, 36,  48,  81,  72,  84, 126, ...
   5, 10, 18, 27, 42,  54, 104, 110,  88, 165, ...
   6, 12, 21, 28, 44,  56, 105, 112,  96, 256, ...
   7, 14, 33, 32, 45,  64, 136, 114, 100, 258, ...
  11, 19, 35, 52, 50,  78, 148, 128, 108, 266, ...
  13, 22, 37, 55, 57,  92, 152, 130, 132, 296, ...
  17, 25, 38, 74, 63,  95, 164, 135, 138, 304, ...
  23, 26, 39, 77, 66,  99, 182, 147, 156, 322, ...
  29, 31, 46, 85, 68, 102, 186, 154, 184, 369, ...

Alois P. Heinz, Antidiagonals n = 1..18, flattened

Columns k=2-16 give: A114842, A114841, A114843, A114840, A114839, A114838, A114837, A114836, A114826, A114825, A114824, A114823, A117529, A117551, A117550.
Row n=1 gives: A060320.
Cf. A000045, A001221, A022307, A303215, A303218.

F:= combinat[fibonacci]: with(numtheory):
A:= proc() local h, p, q; p, q:= proc() [] end, 2;
      proc(n, k)
        while nops(p(k))

nmax = 12; maxIndex = 200;
nu[n_] := nu[n] = PrimeNu[Fibonacci[n]];
col[k_] := Select[Range[maxIndex], nu[#] == k&];
T = Array[col, nmax];
A[n_, k_] := T[[k, n]];
Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 04 2020 *)

A000045(A(n,k)) = A303218(n,k).

A001221(A000045(A(n,k))) = k.

A060319 Smallest Fibonacci number with n distinct prime factors.

Original entry on oeis.org

1, 2, 21, 610, 6765, 832040, 102334155, 190392490709135, 1548008755920, 23416728348467685, 2880067194370816120, 81055900096023504197206408605, 2706074082469569338358691163510069157, 5358359254990966640871840, 57602132235424755886206198685365216, 18547707689471986212190138521399707760

Offset: 0

Labos Elemer, Mar 28 2001

nonn

			a(5) = F(30) = 832040 = 2^3 * 5 * 11 * 41 * 61.

Amiram Eldar, Table of n, a(n) for n = 0..40
Ron Knott, Fibonacci numbers with tables of F(0)-F(500)

Cf. A001605, A005478, A060320.

Row n=1 of A303218.

f[n_]:=Length@FactorInteger[Fibonacci[n]]; lst={};Do[Do[If[f[n]==q,Print[Fibonacci[n]];AppendTo[lst,Fibonacci[n]];Break[]],{n,280}],{q,18}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 23 2009 *)
First /@ SortBy[#, Last] &@ Map[#[[1]] &, Values@ GroupBy[#, Last]] &@ Table[{#, PrimeNu@ #} &@ Fibonacci@ n, {n, 2, 200}] (* Michael De Vlieger, Feb 18 2017, Version 10 *)

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

A060383 a(1) = a(2) = 1; for n >2, a(n) = smallest prime factor of n-th Fibonacci number.

Original entry on oeis.org

1, 1, 2, 3, 5, 2, 13, 3, 2, 5, 89, 2, 233, 13, 2, 3, 1597, 2, 37, 3, 2, 89, 28657, 2, 5, 233, 2, 3, 514229, 2, 557, 3, 2, 1597, 5, 2, 73, 37, 2, 3, 2789, 2, 433494437, 3, 2, 139, 2971215073, 2, 13, 5, 2, 3, 953, 2, 5, 3, 2, 59, 353, 2, 4513, 557, 2, 3, 5, 2, 269, 3, 2, 5

Offset: 1

Labos Elemer, Apr 03 2001

nonn

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

Tyler Busby, Table of n, a(n) for n = 1..1452 (terms 1..1000 from Alois P. Heinz, derived from Kelly's data, terms 1001..1408 from Amiram Eldar)
Blair Kelly, Fibonacci and Lucas Factorizations

Cf. A000045, A022307, A001605, A060319, A060320, A051694, A139044.

[1,1] cat [Minimum(PrimeDivisors(Fibonacci(n))): n in [3..70]]; // Vincenzo Librandi, Dec 25 2016

f[n_] := (FactorInteger@ Fibonacci@ n)[[1,1]]; Array[f, 70] (* Robert G. Wilson v, Jul 07 2007 *)

a(n) = if ((f=fibonacci(n))==1, 1, factor(f)[1,1]); \\ Michel Marcus, Nov 15 2014

a(n) = A020639(A000045(n)). - Michel Marcus, Nov 15 2014

Better definition from Omar E. Pol, Apr 25 2008

A359850 a(n) is the index of the smallest tribonacci number (A000073) with exactly n distinct prime factors.

Original entry on oeis.org

2, 4, 8, 13, 20, 29, 49, 56, 101, 93, 124, 268, 221

Offset: 0

Ilya Gutkovskiy, Jan 15 2023

Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Tribonacci Number

Cf. A000073, A001221, A060320, A359848, A359851.

a(11)-a(12) from Daniel Suteu, Jan 17 2023

A359851 a(n) is the index of the smallest tetranacci number (A000078) with exactly n distinct prime factors.

Original entry on oeis.org

3, 5, 8, 15, 20, 30, 53, 60, 80, 89, 130, 252

Offset: 0

Ilya Gutkovskiy, Jan 15 2023

Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Tetranacci Number

Cf. A000078, A001221, A060320, A359849, A359850.

a(11) from Daniel Suteu, Jan 17 2023

A359853 a(n) is the index of the smallest Fibonacci n-step number with exactly n distinct prime factors.

Original entry on oeis.org

8, 13, 20, 18, 50, 56, 73, 95, 267

Offset: 2

Ilya Gutkovskiy, Jan 15 2023

Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A001221, A060320, A359850, A359851, A359852.

a(10) from Jinyuan Wang, Jan 17 2023

A229491 The index of the smallest Lucas number having exactly n distinct prime factors.

Original entry on oeis.org

1, 0, 6, 12, 30, 42, 45, 75, 126, 105, 175, 135, 255, 195, 285, 345, 435, 315, 693, 405, 525, 765, 585

Offset: 0

T. D. Noe, Oct 28 2013

nonn

Cf. A000032 (Lucas numbers).
Cf. A060320 (corresponding sequence for Fibonacci numbers).
Cf. A086598 (number of distinct factors in Lucas numbers).
Cf. A229490 (smallest Lucas number having exactly n distinct prime factors).

cp's OEIS Frontend

A022307 Number of distinct prime factors of n-th Fibonacci number.

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A060384 Number of decimal digits in n-th Fibonacci number.

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A303217 A(n,k) is the n-th index of a Fibonacci number with exactly k distinct prime factors; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A060319 Smallest Fibonacci number with n distinct prime factors.

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A060385 Largest prime factor of n-th Fibonacci number.

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A060383 a(1) = a(2) = 1; for n >2, a(n) = smallest prime factor of n-th Fibonacci number.

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A359850 a(n) is the index of the smallest tribonacci number (A000073) with exactly n distinct prime factors.

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