

A099267


Numbers generated by the golden sieve.


12



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,...
Positions of 1 in A189479.  Clark Kimberling, Apr 22 2011


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences generated by sieves


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
a(n) = a(a(n))  n.  Marc Morgenegg, Sep 23 2019


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)
 Reinhard Zumkeller, Sep 18 2011


CROSSREFS

Numbers n such that a(n+1)a(n)=2 are given by A004956.
If prefixed by an initial 1, same as A026355.
Cf. A001622, A136119, A007066, A189479.
Complement of A007066.  Gerald Hillier, Dec 19 2008
Cf. A193213 (primes).
Sequence in context: A260396 A029921 A026355 * A007067 A186322 A092979
Adjacent sequences: A099264 A099265 A099266 * A099268 A099269 A099270


KEYWORD

nonn,easy,nice


AUTHOR

Benoit Cloitre, Nov 15 2002


STATUS

approved



