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A280104
a(n) = smallest prime factor of n-th Lucas number A000032(n), or 1 if there are none.
2
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)
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
KEYWORD
nonn
AUTHOR
Vincenzo Librandi, Dec 26 2016
EXTENSIONS
Offset changed from Bruno Berselli, Dec 27 2016
STATUS
approved