

A099267


Numbers generated by the golden sieve.


13



2, 3, 5, 6, 8, 10, 11, 13, 14, 16, 18, 19, 21, 23, 24, 26, 27, 29, 31, 32, 34, 35, 37, 39, 40, 42, 44, 45, 47, 48, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 73, 74, 76, 78, 79, 81, 82, 84, 86, 87, 89, 90, 92, 94, 95, 97, 99, 100, 102, 103, 105, 107, 108, 110
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OFFSET

1,1


COMMENTS

Let f(n) denote the nth term of the current working sequence. Start with the positive integers:
1,2,3,4,5,6,7,8,9,10,11,12,...
Delete the term in position f(1), which is f(f(1))=f(1)=1, leaving:
2,3,4,5,6,7,8,9,10,11,12,...
Delete the term in position f(2), which is f(f(2))=f(3)=4, leaving:
2,3,5,6,7,8,9,10,11,12,...
Delete the term in position f(3), which is f(f(3))=f(5)=7, leaving:
2,3,5,6,8,9,10,11,12,...
Delete the term in position f(4), which is f(f(4))=f(6)=9, leaving:
2,3,5,6,8,10,11,12,...
Iterating the "sieve" indefinitely produces the sequence:
2,3,5,6,8,10,11,13,14,16,18,19,21,23,24,26,27,29,31,32,34,35,37,39,...


LINKS



FORMULA

a(n) = floor(n*phi + 2  phi) where phi = (1 + sqrt(5))/2.
a(a(...a(1)...)) with n iterations equals F(n+1) = A000045(n+1).
For n>0 and k>0 we have a(a(n) + F(k)  (1 + (1)^k)/2) = a(a(n)) + F(k+1)  1  (1)^k.  Benoit Cloitre, Nov 22 2004


MATHEMATICA

t = Nest[Flatten[# /. {0 > {0, 1}, 1 > {1, 0, 1}}] &, {0}, 6] (*A189479*)
Flatten[Position[t, 0]] (*A007066*)
Flatten[Position[t, 1]] (*A099267*)


PROG

(Haskell)
a099267 n = a099267_list !! (n1)
a099267_list = f 1 [1..] 0 where
f k xs y = ys' ++ f (k+1) (ys ++ xs') g where
ys' = dropWhile (< y) ys
(ys, _:xs') = span (< g) xs
g = xs !! (h  1)
h = xs !! (k  1)


CROSSREFS

Numbers n such that a(n+1)a(n)=2 are given by A004956.
If prefixed by an initial 1, same as A026355.


KEYWORD

nonn,easy,nice


AUTHOR



STATUS

approved



