A115051 - OEIS

%I #20 Feb 12 2020 06:08:57

%S 1,1,1,3,4,5,4,4,6,15,4,9,3,8,22,42,61

%N Number of distinct prime factors of F(n + L(n)) where F(n) is the Fibonacci number and L(n) is the Lucas number and n >= 2.

%C Added a(13)=9 from F(534) and a(14)=3 from F(857) using Kelly's factorizations. a(15)>=5 via F(1379) and a(16)=22 via F(2223). - _R. J. Mathar_, Apr 23 2006

%C a(19) >= 24. - _Amiram Eldar_, Feb 12 2020

%H Blair Kelley, <a href="http://mersennus.net/fibonacci//">Factorizations</a>

%H FactorDB, <a href="http://factordb.com/index.php?query=I9368">Status of F(9368)</a>.

%e The first three terms are 1 since:

%e F(2 + L(2)) = 5 (a prime)

%e F(3 + L(3)) = 13 (a prime)

%e F(4 + L(4)) = 89 (a prime)

%p lucas := proc(n::integer) if n = 0 then RETURN(2) ; elif n = 1 then RETURN(1) ; else RETURN(combinat[fibonacci](n-1)+combinat[fibonacci](n+1)) ; fi ; end : for n from 2 to 100 do print(n+lucas(n),"...") ; tst := combinat[fibonacci](n+lucas(n)) ; an := nops(op(2,ifactors(tst))) ; print(an) ; od : # _R. J. Mathar_, Apr 23 2006

%t Table[PrimeNu[Fibonacci[n+LucasL[n]]],{n,2,15}] (* _Harvey P. Dale_, Nov 12 2016 *)

%Y Cf. A000045, A000032.

%K nonn,more

%O 2,4

%A _Parthasarathy Nambi_, Feb 28 2006

%E More terms from _R. J. Mathar_, Apr 23 2006

%E Offset corrected and a(15)-a(18) added from factordb.com by _Amiram Eldar_, Feb 12 2020