A280104 a(n) = smallest prime factor of n-th Lucas number A000032(n), or 1 if there are none.

2, 1, 3, 2, 7, 11, 2, 29, 47, 2, 3, 199, 2, 521, 3, 2, 2207, 3571, 2, 9349, 7, 2, 3, 139, 2, 11, 3, 2, 7, 59, 2, 3010349, 1087, 2, 3, 11, 2, 54018521, 3, 2, 47, 370248451, 2, 6709, 7, 2, 3, 6643838879, 2, 29, 3, 2, 7, 119218851371, 2, 11, 47, 2, 3, 709, 2

Offset: 0

Author

Vincenzo Librandi, Dec 26 2016

Keywords

• nonn

Comments

From Robert Israel, Jan 05 2017: (Start)

If m and n are odd, m > 1 and m | n, then a(n) <= a(m).

a(n) = 2 if and only if 3 | n.

a(n) = 3 if and only if n is in A091999.

a(n) is never 5.

a(n) = 7 if and only if n is in A259755.

a(n) = A000032(n) if and only if n is in A001606.

(End)

Links

• Robert Israel, Table of n, a(n) for n = 0..1000
• Fibonacci and Lucas Factorizations

Crossrefs

Cf. A000032, A001606, A020639, A079451 (same for largest prime factor), A091999, A139044, A144293, A259755, A279623.

Column k=2 of A238899 (for n>=2).

Programs

Magma

[2,1] cat [Minimum(PrimeDivisors(Lucas(n))): n in [2..60]];

Maple

lucas:= n -> combinat:-fibonacci(n+1)+combinat:-fibonacci(n-1):
spf:= proc(n) local F;
  F:= remove(hastype,ifactors(n,easy)[2],symbol);
  if F <> [] then return min(seq(f[1],f=F)) fi;
min(numtheory:-factorsec(n))
end proc:
spf(1):= 1:
map(spf @ lucas, [$0..200]); # Robert Israel, Jan 05 2017

Mathematica

f[n_]:=(FactorInteger@LucasL@n)[[1, 1]]; Array[f, 60, 0]

PARI

a000032(n) = fibonacci(n+1)+fibonacci(n-1)
a(n) = if(a000032(n-1)==1, 1, factor(a000032(n-1))[1, 1]) \\ Felix Fröhlich, Dec 26 2016

Formula

a(n) = A020639(A000032(n)). - Felix Fröhlich, Dec 26 2016

Extensions

Offset changed from Bruno Berselli, Dec 27 2016