A060442 -id:A060442 - 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

A080648 Sum of prime factors of Fibonacci(n).

Original entry on oeis.org

0, 0, 2, 3, 5, 2, 13, 10, 19, 16, 89, 5, 233, 42, 68, 57, 1597, 38, 150, 60, 436, 288, 28657, 35, 3006, 754, 181, 326, 514229, 110, 2974, 2264, 19892, 5168, 141979, 148, 2443, 9499, 135956, 2228, 62158, 676, 433494437, 641, 109526, 29257, 2971215073, 1185

Offset: 1

Joseph L. Pe, Feb 28 2003

nonn

			a(8) = 10 because Fibonacci(8) = 21 and the sum of the prime divisors {3, 7} equals 10.

Amiram Eldar, Table of n, a(n) for n = 1..1408 (terms 1..1000 from T. D. Noe, using Blair Kelly's data)
Blair Kelly, Fibonacci and Lucas Factorizations

Cf. A000045, A008472, A060442 (Fibonacci prime factors).

[&+PrimeDivisors(Fibonacci(n)):n in [1..48]]; // Marius A. Burtea, Oct 15 2019

with (numtheory):with(combinat, fibonacci):
sopf:= proc(n) local e, j; e := ifactors(fibonacci(n))[2]:
add (e[j][1], j=1..nops(e)) end:
seq (sopf(n), n=1..100); # Michel Lagneau, Nov 13 2012
A080648 := proc(n)
    A008472(combinat[fibonacci](n)) ;
end proc: # R. J. Mathar, Nov 15 2012
# third Maple program:
a:= n-> add(i[1], i=ifactors((<<0|1>, <1|1>>^n)[1, 2])[2]):
seq(a(n), n=1..48);  # Alois P. Heinz, Sep 03 2019

Table[Apply[Plus, Transpose[FactorInteger[Fibonacci[n]]][[1]]], {n, 3, 100}] (* Pe *)
Array[Plus@@First/@FactorInteger[Fibonacci[ # ]]&, 40 ] (* Michel Lagneau, Nov 13 2012 *)

a(n) = vecsum(factor(fibonacci(n))[,1]); \\ Michel Marcus, Oct 15 2019

A233281 Numbers n such that the least Fibonacci number F_k which is a multiple of n has a prime index, i.e., k is in A000040.

Original entry on oeis.org

2, 5, 13, 37, 73, 89, 113, 149, 157, 193, 233, 269, 277, 313, 353, 389, 397, 457, 557, 613, 673, 677, 733, 757, 877, 953, 977, 997, 1069, 1093, 1153, 1213, 1237, 1453, 1597, 1657, 1753, 1873, 1877, 1933, 1949, 1993, 2017, 2137, 2221, 2237, 2309, 2333, 2417, 2473

Offset: 1

Antti Karttunen, Dec 13 2013

nonn

Numbers n such that A001177(n) is prime.
Each natural number n belongs to this sequence if the smallest Fibonacci number which it divides is a term of A030426. - Jon E. Schoenfield, Feb 28 2014
A092395 gives all the primes in this sequence (cf. Wikipedia-link), and the first composite occurs as the 69th term, where a(69)=4181 while A092395(69)=4273. After 4181 (= 37*113 = F_19), the next term missing from A092395 is a(148)=10877 (= 73*149. A001177(10877) = 37, F_37 = 24157817 = 2221*10877). Both of these numbers (4181 and 10877) occur in various lists of Fibonacci-related pseudoprimes. Sequence A238082 gives all composites occurring in this sequence.
If n is in this sequence then all divisors d > 1 of n are in this sequence. - Charles R Greathouse IV, Feb 04 2014
Composite members begin 4181, 10877, 75077, 162133, 330929, .... - Charles R Greathouse IV, Mar 07 2014

Antti Karttunen and Charles R Greathouse IV, Table of n, a(n) for n = 1..2000 (first 157 terms from Karttunen)
Wikipedia, Fibonacci prime, section: Divisibility of Fibonacci numbers

Disjoint union of A092395 and A238082. The first 68 terms are identical with A092395, after which follows the first case of the latter sequence, with a(69) = A238082(1) = 4181.

Cf. A000045, A001177, A001602, A030426, A051694, A060442, A086597, A233282.

a233281 n = a233281_list !! (n-1)
a233281_list = filter ((== 1) . a010051 . a001177) [1..]
-- Reinhard Zumkeller, Apr 04 2014

is(n)=my(k); while(fibonacci(k++)%n, ); isprime(k) \\ Charles R Greathouse IV, Feb 04 2014

entry(p)=my(k=1);while(fibonacci(k++)%p,);k;
is(n)={
    if(n%2==0,return(n==2));
    if(n<13, return(n==5));
    my(f=factor(n),p,F);
    if(f[1,2]>1 && f[1,1]<1e14,return(0));
    p=entry(f[1,1]);
    F=fibonacci(p);
    if(f[1,2]>1 && F%f[1,1]^f[1,2],return(0));
    if(!isprime(p), return(0));
    for(i=2,#f~,
        if(F%f[i,1]^f[i,2],return(0))
    );
    1
}; \\ Charles R Greathouse IV, Feb 04 2014

A010051(A001177(a(n))) = 1. - Reinhard Zumkeller, Apr 04 2014

A060441 Triangle T(n,k), n >= 0, in which n-th row (for n >= 3) lists prime factors of Fibonacci(n) (see A000045), with repetition.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 2, 2, 2, 13, 3, 7, 2, 17, 5, 11, 89, 2, 2, 2, 2, 3, 3, 233, 13, 29, 2, 5, 61, 3, 7, 47, 1597, 2, 2, 2, 17, 19, 37, 113, 3, 5, 11, 41, 2, 13, 421, 89, 199, 28657, 2, 2, 2, 2, 2, 3, 3, 7, 23, 5, 5, 3001, 233, 521, 2, 17, 53, 109, 3, 13, 29, 281, 514229, 2, 2, 2, 5, 11, 31, 61

Offset: 0

N. J. A. Sloane, Apr 07 2001

Rows have irregular lengths.

T(n,k) = A027746(A000045(n),k), k = 1 .. A038575(n). - Reinhard Zumkeller, Aug 30 2014

			Triangle begins:
  0;
  1;
  1;
  2;
  3;
  5;
  2, 2, 2;
  13;
  3, 7;
  2, 17;
  ...

Blair Kelly, Fibonacci and Lucas factorizations

A000045, A060442.

Cf. A038575 (row lengths), A027746, A001222.

a060441 n k = a060441_tabf !! (n-1) !! (k-1)
a060441_row n = a060441_tabf !! (n-1)
a060441_tabf = [0] : [1] : [1] : map a027746_row (drop 3 a000045_list)
-- Reinhard Zumkeller, Aug 30 2014

with(combinat); A060441 := n->ifactor(fibonacci(n));
with(numtheory): with(combinat): for i from 3 to 50 do for j from 1 to nops(ifactors(fibonacci(i))[2]) do for k from 1 to ifactors(fibonacci(i))[2][j][2] do printf(`%d,`, ifactors(fibonacci(i))[2][j][1]) od: od: od:

More terms from James Sellers, Apr 09 2001

A193615 Second-largest prime factor of the n-th Fibonacci number, if composite, or 1 otherwise.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 5, 1, 3, 1, 13, 5, 7, 1, 17, 37, 11, 13, 89, 1, 7, 5, 233, 53, 29, 1, 31, 557, 47, 89, 1597, 13, 19, 149, 113, 233, 41, 2789, 211, 1, 199, 61, 461, 1, 47, 97, 151, 1597, 521, 953, 109, 661, 281, 797, 19489, 353, 61, 4513

Offset: 3

Charles R Greathouse IV, Jul 31 2011

nonn

For clarification: if the largest prime factor occurs more than once, then that prime factor is selected.

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

Charles R Greathouse IV, Table of n, a(n) for n = 3..1422

Cf. A060385, A061488, A060442.

factors[n_] := Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]]; fn[n_] := Module[{fibn = Fibonacci[n]}, If[PrimeQ[fibn], 1, factors[fibn][[-2]]]]; Table[fn[n], {n, 3, 80}]

a(n)=my(f=factor(fibonacci(n)),t=#f[,1]);if(f[t,2]==1,if(t==1,1,f[t-1,1]),f[t,1])

A131401 Least number dividing Fibonacci(n) but not dividing Fibonacci(m) for m < n, or 0 if there is no such number.

Original entry on oeis.org

1, 0, 2, 3, 5, 4, 13, 7, 17, 11, 89, 6, 233, 29, 10, 47, 1597, 19, 37, 15, 26, 199, 28657, 14, 25, 521, 53, 39, 514229, 20, 557, 2207, 178, 3571, 65, 27, 73, 9349, 466, 35, 2789, 52, 433494437, 43, 85, 139, 2971215073, 64, 97, 101, 3194, 699, 953, 212, 445, 49, 74, 59

Offset: 1

Herbert A. Hauptman (hauptman(AT)hwi.buffalo.edu) & Robert G. Wilson v, Jul 07 2007

nonn

First occurrence of n in A001177 or 0 if impossible.
Conjecture: only a(2)=0. I have not found values of a(n) < 2*106 less than 100 for n = 43, 47, 74, 82, 83 & 94.
When Fibonacci(n) is a prime number, then a(n)=Fibonacci(n). Note that a(n)=0 for n=2 because Fibonacci(1)=Fibonacci(2)=1. For n > 2, an upper bound for a(n) is Fibonacci(n). The difficulty in computing this sequence for large n is the factorization of Fibonacci(n), which is required to find the divisors of Fibonacci(n). - T. D. Noe, Jan 12 2009
In other words, the conjecture is true. For n > 2, Fibonacci(n) has at least one divisor that does not divide Fibonacci(k) for k < n. The number of such divisors is A120256(n).

Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers, Afterword by Herbert A. Hauptman, 2. 'The Minor Modulus m(n)', Prometheus Books, NY, 2007, pp. 329-342.

T. D. Noe, Table of n, a(n) for n = 1..300

Cf. A060442. - T. D. Noe, Jan 12 2009

f[n_] := Block[{k = 1}, While[Mod[Fibonacci@k, n] != 0 && k < 101, k++ ]; k]; t = Table[0, {100}]; Do[ a = f@n; If[a < 101 && t[[a]] == 0, t[[a]] = n; Print[{a, n}]], {n, 106}]
nn=100; fib=Fibonacci[Range[nn]]; Join[{1,0}, Table[dvrs=Rest[Divisors[fib[[n]]]]; k=1; While[d=dvrs[[k]]; pos=Position[fib,?(Mod[ #,d]==0&),1,1]; pos!={{n}}, k++ ]; d, {n,3,nn}]] (* _T. D. Noe, Jan 12 2009 *)

Extended by T. D. Noe, Jan 12 2009

cp's OEIS Frontend

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

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A080648 Sum of prime factors of Fibonacci(n).

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A233281 Numbers n such that the least Fibonacci number F_k which is a multiple of n has a prime index, i.e., k is in A000040.

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A060441 Triangle T(n,k), n >= 0, in which n-th row (for n >= 3) lists prime factors of Fibonacci(n) (see A000045), with repetition.

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A193615 Second-largest prime factor of the n-th Fibonacci number, if composite, or 1 otherwise.

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A131401 Least number dividing Fibonacci(n) but not dividing Fibonacci(m) for m < n, or 0 if there is no such number.

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