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A057854 Non-Lucas numbers: the complement of A000032.

%I A057854 #36 Sep 11 2024 00:32:06
%S A057854 5,6,8,9,10,12,13,14,15,16,17,19,20,21,22,23,24,25,26,27,28,30,31,32,
%T A057854 33,34,35,36,37,38,39,40,41,42,43,44,45,46,48,49,50,51,52,53,54,55,56,
%U A057854 57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,77,78,79,80
%N A057854 Non-Lucas numbers: the complement of A000032.
%C A057854 The formula is a consequence of the Lambek-Moser theorem.
%H A057854 G. C. Greubel, <a href="/A057854/b057854.txt">Table of n, a(n) for n = 1..10000</a>
%H A057854 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lambek%E2%80%93Moser_theorem">Lambek-Moser theorem</a>
%F A057854 a(n) = floor(1/2 - LambertW(-1, -log(phi)/phi^(n+1/2))/log(phi)) with phi = (1+sqrt(5))/2. - Nicolas Normand (nicolas.normand (at) polytech.univ-nantes.fr)
%F A057854 a(n) = A090946(n+2). - _R. J. Mathar_, Jan 29 2019
%p A057854 a := proc(n) floor(-1/ln(1/2+1/2*5^(1/2))*LambertW(-1,-ln(1/2+1/2*5^(1/2))/ ((1/2+1/2*5^(1/2))^(n+1/2)))+1/2) end; # _Simon Plouffe_, Nov 30 2017
%p A057854 # alternative
%p A057854 isA000032 := proc(n)
%p A057854     local l1,l2 ;
%p A057854     if n <= 0 then
%p A057854         false;
%p A057854     elif n <= 4 then
%p A057854         true ;
%p A057854     else
%p A057854         l1 := 3 ; l2 := 4 ;
%p A057854         while true do
%p A057854             l := l1+l2 ;
%p A057854             if l > n then
%p A057854                 return false;
%p A057854             elif l = n then
%p A057854                 return true;
%p A057854             else
%p A057854                 l1 := l2 ; l2 := l ;
%p A057854             end if;
%p A057854         end do:
%p A057854     end if;
%p A057854 end proc:
%p A057854 isA057854 := proc(n)
%p A057854     not isA000032(n) ;
%p A057854 end proc:
%p A057854 A057854 := proc(n)
%p A057854     option remember;
%p A057854     if n = 1 then
%p A057854         5 ;
%p A057854     else
%p A057854         for a from procname(n-1)+1 do
%p A057854             if isA057854(a) then
%p A057854                 return a;
%p A057854             end if;
%p A057854         end do:
%p A057854     end if;
%p A057854 end proc:
%p A057854 seq(A057854(n),n=1..10) ; # _R. J. Mathar_, Feb 01 2019
%t A057854 a[n_] := With[{phi = (1 + Sqrt[5])/2}, Floor[1/2 - LambertW[-1, -Log[phi]/phi^(n + 1/2)]/Log[phi]]];
%t A057854 Table[a[n], {n, 1, 70}] (* _Peter Luschny_, Nov 30 2017 *)
%t A057854 b:= Complement[Range[1, 100], LucasL[Range[20]]]; Table[b[[n+1]], {n, 1, 70}] (* _G. C. Greubel_, Jun 19 2019 *)
%o A057854 (Python)
%o A057854 def A057854(n):
%o A057854     def f(x):
%o A057854         if x<=2: return n+2
%o A057854         a, b, c = 1, 3, 0
%o A057854         while b<=x:
%o A057854             a, b = b, a+b
%o A057854             c += 1
%o A057854         return n+c+2
%o A057854     m, k = n, f(n)
%o A057854     while m != k: m, k = k, f(k)
%o A057854     return m # _Chai Wah Wu_, Sep 10 2024
%K A057854 nonn,easy
%O A057854 1,1
%A A057854 _Roger Cuculière_, Nov 12 2000
%E A057854 More terms from Larry Reeves (larryr(AT)acm.org), Nov 28 2000