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

Author

Benoit Cloitre, Nov 15 2002

Keywords

• nonn
• easy
• nice

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
• Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 3.
• Index entries for sequences generated by sieves

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).

Programs

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

Mathematica

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

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