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%I #9 May 30 2017 05:36:35
%S 1,4,18,5778,215002084978043708894524818,
%T 8000328475168735073037785452636987975637751878418,
%U 267093222236137978360266538108484045754096036229865700498,8916982544642128998138920801180413422215946187628307595501392018
%N Practical Lucas numbers.
%C Melfi proved that this sequence is infinite.
%C The indices of these Lucas numbers are 1, 3, 6, 18, 126, 234, 270, 306, 342, 378, 450, 522 ...
%H Amiram Eldar, <a href="/A287677/b287677.txt">Table of n, a(n) for n = 1..12</a>
%H Giuseppe Melfi, <a href="http://members.unine.ch/giuseppe.melfi/articoli/smapoto.pdf">A survey on practical numbers</a>, Rend. Sem. Mat. Univ. Pol. Torino, 53, (1995), 347-359.
%e 18 is in this sequence since it is the 6th Lucas number, A000032(6) and it is also a practical number, A005153(8).
%t PracticalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1 || (n>1 && OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e} = Transpose[f]; Do[If[p[[i]] > 1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]]; Select[Table[LucasL[n],{n,1,310}], PracticalQ]
%Y Cf. A000032, A005153, A124105, A287679.
%K nonn
%O 1,2
%A _Amiram Eldar_, May 29 2017