A099267 Numbers generated by the golden sieve.

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

Offset

1,1

Comments

Let f(n) denote the n-th 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 !! (n-1)
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