A057854 Non-Lucas numbers: the complement of A000032.

5, 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80

Offset

1,1

Comments

The formula is a consequence of the Lambek-Moser theorem.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Wikipedia, Lambek-Moser theorem

Formula

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)

a(n) = A090946(n+2). - R. J. Mathar, Jan 29 2019

Maple

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
# alternative
isA000032 := proc(n)
    local l1, l2 ;
    if n <= 0 then
        false;
    elif n <= 4 then
        true ;
    else
        l1 := 3 ; l2 := 4 ;
        while true do
            l := l1+l2 ;
            if l > n then
                return false;
            elif l = n then
                return true;
            else
                l1 := l2 ; l2 := l ;
            end if;
        end do:
    end if;
end proc:
isA057854 := proc(n)
    not isA000032(n) ;
end proc:
A057854 := proc(n)
    option remember;
    if n = 1 then
        5 ;
    else
        for a from procname(n-1)+1 do
            if isA057854(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A057854(n), n=1..10) ; # R. J. Mathar, Feb 01 2019

Mathematica

a[n_] := With[{phi = (1 + Sqrt[5])/2}, Floor[1/2 - LambertW[-1, -Log[phi]/phi^(n + 1/2)]/Log[phi]]];
Table[a[n], {n, 1, 70}] (* Peter Luschny, Nov 30 2017 *)
b:= Complement[Range[1, 100], LucasL[Range[20]]]; Table[b[[n+1]], {n, 1, 70}] (* G. C. Greubel, Jun 19 2019 *)

Prog

(Python)
def A057854(n):
    def f(x):
        if x<=2: return n+2
        a, b, c = 1, 3, 0
        while b<=x:
            a, b = b, a+b
            c += 1
        return n+c+2
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Sep 10 2024

Crossrefs

Sequence in context: A201512 A242290 A296562 * A033266 A355641 A363701

Adjacent sequences: A057851 A057852 A057853 * A057855 A057856 A057857

Keyword

nonn,easy

Author

Roger Cuculière, Nov 12 2000

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Nov 28 2000

Status

approved