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A099267
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Numbers generated by the golden sieve.
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9
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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; internal format)
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OFFSET
| 1,1
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COMMENTS
| Let f(n) denote the n-th term of the current working sequence. Start with the natural 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,...
The subsequence of primes in this sequence begins: 2, 3, 5, 11, 13, 19, 23, 29, 31, 37, 47, 53, 61, 71, 73, 79, 89, 97, 103, 107. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 25 2010]
Positions of 1 in A189479. [From Clark Kimberling, Apr 22 2011]
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LINKS
| Index entries for sequences generated by sieves
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
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FORMULA
| a(n) = floor(n*phi+2-phi) where phi=(1+sqrt(5))/2. Also 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 (benoit7848c(AT)orange.fr), Nov 22 2004
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MATHEMATICA
| t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0, 1}}] &, {0}, 6] (*A189479*)
Flatten[Position[t, 0]] (*A007066*)
Flatten[Position[t, 1]] (*A099267*)
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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
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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. A136119.
Complement of A007066 [From Gerald Hillier (adr.rabbicat(AT)gmail.com), Dec 19 2008]
Cf. A193213 (primes).
Sequence in context: A047448 A029921 A026355 * A007067 A186322 A092979
Adjacent sequences: A099264 A099265 A099266 * A099268 A099269 A099270
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 15 2002
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