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A001578 Smallest primitive prime factor of Fibonacci number F(n), or 1 if F(n) has no primitive prime factor.
(Formerly M0603 N0217)
13
1, 1, 2, 3, 5, 1, 13, 7, 17, 11, 89, 1, 233, 29, 61, 47, 1597, 19, 37, 41, 421, 199, 28657, 23, 3001, 521, 53, 281, 514229, 31, 557, 2207, 19801, 3571, 141961, 107, 73, 9349, 135721, 2161, 2789, 211, 433494437, 43, 109441, 139, 2971215073, 1103, 97, 101 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

A prime factor of F(n) is called primitive if it does not divide F(r) for any r < n.

A Fibonacci number can have more than one primitive factor; the primitive factors of F(19) are 37 and 113.

From Robert Israel, Oct 13 2015: (Start)

Since gcd(F(n),F(k)) = F(gcd(n,k)), the non-primitive prime factors of F(n) are factors of F(k) for some proper divisors k of n.

Since prime p divides F(p-1) if p == 1 or 4 (mod 5), F(p+1) if p == 2 or 3 mod 5, F(p) if p = 5, we have a(n) >= n-1 if a(n) > 1.

a(n) = n-1 iff n=2 or n-1 is in A000057.

a(n) = n+1 iff n+1 is a prime in A106535. (End)

REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000 (using Blair Kelly's factorizations)

D. Jarden, On the greatest primitive divisors of Fibonacci and Lucas numbers with prime-power subscripts, Fib. Quart. 1(#3) (1963), 15-31.

Blair Kelly, Fibonacci and Lucas Factorizations

MAPLE

for n from 1 to 350 do

  f:= combinat:-fibonacci(n);

  if not isprime(n) then

    for k in map(t -> n/t, numtheory:-factorset(n)) do

       fk:= combinat:-fibonacci(k);

       g:= igcd(f, fk);

       while g > 1 do

         f:= f/g;

         g:= igcd(f, fk);

       od

    od

  fi;

  if f = 1 then A[n]:= 1; next fi;

  F:= map(t -> t[1], ifactors(f, easy)[2]);

  p:= select(type, F, integer);

  if nops(p) >= 1 then A[n]:= min(p); next fi;

  A[n]:= min(numtheory:-factorset(f));

od:

seq(A[i], i=1..350); # Robert Israel, Oct 13 2015

MATHEMATICA

prms={}; Table[f=First/@FactorInteger[Fibonacci[n]]; p=Complement[f, prms]; prms=Join[prms, p]; If[p=={}, 1, First[p]], {n, 50}]

CROSSREFS

Cf. A000045, A000057, A106535, A086597 (number of primitive prime factors in F(n)), A061488 (1's omitted), A262341 (largest primitive prime factor of F(n)).

Sequence in context: A276350 A262215 A030790 * A262341 A178763 A111141

Adjacent sequences:  A001575 A001576 A001577 * A001579 A001580 A001581

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited by T. D. Noe, Apr 15 2004

Definition clarified at the suggestion of Joerg Arndt by Jonathan Sondow, Oct 13 2015

STATUS

approved

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Last modified May 29 10:48 EDT 2017. Contains 287246 sequences.