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A084385
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a(1) = 1; a(n+1) is the smallest number not occurring earlier and coprime to Sum_{j=1..n} a(j).
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4
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1, 2, 4, 3, 7, 5, 9, 6, 8, 11, 13, 10, 12, 15, 17, 14, 16, 19, 21, 18, 20, 23, 25, 22, 24, 27, 29, 26, 28, 31, 33, 30, 32, 35, 37, 34, 36, 39, 41, 38, 40, 43, 45, 42, 44, 47, 49, 46, 48, 51, 53, 50, 52, 55, 57, 54, 56, 59, 61, 58, 60, 63, 65, 62, 64, 67, 69, 66, 68, 71, 73, 70
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OFFSET
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1,2
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COMMENTS
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Rearrangement of the positive integers.
Any sequence defined in this manner (that is, a(1) is any positive integer and a(n+1) is the smallest integer not occurring earlier and coprime to Sum_{j=1..n} a(j)) is a rearrangement of all positive integers. This property is used by problem 4 of Chinese High School Mathematical Olympiad in 2018. - Shu Shang, Sep 29 2021
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LINKS
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FORMULA
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For n > 6: a(n) = n-2 for n mod 4 = 0, a(n) = n-1 for n mod 4 = 1, a(n) = n+1 for n mod 4 = 2, a(n) = n+2 for n mod 4 = 3. - Klaus Brockhaus, Nov 30 2003
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EXAMPLE
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1+2+4 = 7, 3 is the smallest number not occurring earlier and coprime to 7, hence a(4) = 3.
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PROG
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(PARI) used(k, v)=b=0; j=1; while(b<1&&j<=length(v), if(v[j]==k, b=1, j++)); b
{print1(s=1, ", "); v=[s]; for(n=1, 72, j=1; k=2; while(used(k, v)||gcd(k, s)>1, k++); v=concat(v, k); s=s+k; print1(k, ", "))}
(PARI) {print1(1, ", ", 2, ", ", 4, ", ", 3, ", ", 7, ", ", 5, ", "); for(n=7, 73, m=n%4; d=(if(m==0, -2, if(m==1, -1, if(m==2, 1, 2)))); print1(n+d, ", "))}
(Haskell)
import Data.List (delete)
a084385 n = a084385_list !! (n-1)
a084385_list = 1 : f [2..] 1 where
f xs s = g xs where
g (y:ys) = if gcd s y == 1 then y : f (delete y xs) (s + y) else g ys
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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