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

%I A280104 #33 Aug 04 2025 18:43:37
%S A280104 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,
%T A280104 11,3,2,7,59,2,3010349,1087,2,3,11,2,54018521,3,2,47,370248451,2,6709,
%U A280104 7,2,3,6643838879,2,29,3,2,7,119218851371,2,11,47,2,3,709,2
%N A280104 a(n) = smallest prime factor of n-th Lucas number A000032(n), or 1 if there are none.
%C A280104 From _Robert Israel_, Jan 05 2017: (Start)
%C A280104 If m and n are odd, m > 1 and m | n, then a(n) <= a(m).
%C A280104 a(n) = 2 if and only if 3 | n.
%C A280104 a(n) = 3 if and only if n is in A091999.
%C A280104 a(n) is never 5.
%C A280104 a(n) = 7 if and only if n is in A259755.
%C A280104 a(n) = A000032(n) if and only if n is in A001606.
%C A280104 (End)
%H A280104 Robert Israel, <a href="/A280104/b280104.txt">Table of n, a(n) for n = 0..1000</a>
%H A280104 <a href="http://mersennus.net/fibonacci/">Fibonacci and Lucas Factorizations</a>
%F A280104 a(n) = A020639(A000032(n)). - _Felix Fröhlich_, Dec 26 2016
%p A280104 lucas:= n -> combinat:-fibonacci(n+1)+combinat:-fibonacci(n-1):
%p A280104 spf:= proc(n) local F;
%p A280104   F:= remove(hastype,ifactors(n,easy)[2],symbol);
%p A280104   if F <> [] then return min(seq(f[1],f=F)) fi;
%p A280104 min(numtheory:-factorsec(n))
%p A280104 end proc:
%p A280104 spf(1):= 1:
%p A280104 map(spf @ lucas, [$0..200]); # _Robert Israel_, Jan 05 2017
%t A280104 f[n_]:=(FactorInteger@LucasL@n)[[1, 1]]; Array[f, 60, 0]
%o A280104 (Magma) [2,1] cat [Minimum(PrimeDivisors(Lucas(n))): n in [2..60]];
%o A280104 (PARI) a000032(n) = fibonacci(n+1)+fibonacci(n-1)
%o A280104 a(n) = if(a000032(n-1)==1, 1, factor(a000032(n-1))[1, 1]) \\ _Felix Fröhlich_, Dec 26 2016
%Y A280104 Cf. A000032, A001606, A020639, A079451 (same for largest prime factor), A091999, A139044, A144293, A259755, A279623.
%Y A280104 Column k=2 of A238899 (for n>=2).
%K A280104 nonn
%O A280104 0,1
%A A280104 _Vincenzo Librandi_, Dec 26 2016
%E A280104 Offset changed from _Bruno Berselli_, Dec 27 2016