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

0,1

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

Formula

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

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]

Prog

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

Crossrefs

Cf. A000032, A001606, A020639, A079451, A091999, A139044, A144293, A259755, A279623.

Sequence in context: A005247 A363550 A355715 * A135259 A122147 A352418

Adjacent sequences: A280101 A280102 A280103 * A280105 A280106 A280107

Keyword

nonn

Author

Vincenzo Librandi, Dec 26 2016

Extensions

Offset changed from Bruno Berselli, Dec 27 2016

Status

approved