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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 (list; graph; refs; listen; history; text; internal format)
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
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.
Complement of A007066. - Gerald Hillier, Dec 19 2008
Cf. A193213 (primes).
Sequence in context: A260396 A029921 A026355 * A007067 A186322 A092979
KEYWORD
nonn,easy,nice
AUTHOR
Benoit Cloitre, Nov 15 2002
STATUS
approved

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Last modified March 4 13:38 EST 2024. Contains 370532 sequences. (Running on oeis4.)