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A352849
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a(n) is the least k not already in the sequence such that k is pairwise coprime to a(n-1) and a(n-2), starting with a(1) = 1, a(2) = 3, and a(3) = 5.
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1
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1, 3, 5, 2, 7, 9, 4, 11, 13, 6, 17, 19, 8, 15, 23, 14, 25, 27, 16, 29, 21, 10, 31, 33, 20, 37, 39, 22, 35, 41, 12, 43, 47, 18, 49, 53, 24, 55, 59, 26, 45, 61, 28, 51, 65, 32, 57, 67, 34, 63, 71, 38, 69, 73, 40, 77, 79, 30, 83, 89, 36, 85, 91, 44, 75, 97, 46, 81
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OFFSET
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1,2
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COMMENTS
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Sequence begins with 1 and the first 2 odd primes.
a(3k+1) is even for k > 0 as consequence of definition and since 2 is the smallest prime and numbers are either even or odd. Unlike A085229, even numbers in this sequence do not appear in order. Hence a(3k) and a(3k+2) are odd.
The smallest missing number u is even, and there is a smallest missing odd number v that applies to a(3k) and a(3k+2).
Let q be odd and prime. In a given interval i <= n <= j, we either have 2q | a(n) or we have q | a(n) odd.
Regarding 6 | a(n), there are phases i <= n <= j where 6 | a(3k+1) and no a(n) mod 6 = 3 appear. These begin when 3 | a(3k+1) and prevent the entry of 3 | v. Whereupon all u such that 6 | u have been consumed, 3 | a(3k+r), r != 1 occurs, and we move into a phase where we have a(n) mod 6 = 3, but no 6 | a(3k+1) appear. This occurs until all v such that 3 | v have been consumed, and 3 | a(3k+1) once again.
Conjecture: the sequence is a permutation of the natural numbers.
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LINKS
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Michael De Vlieger, Annotated log-log scatterplot of a(n), n = 1..2^12, showing odd numbers congruent to 1 or 5 (mod 6) in green, odd numbers congruent to 3 (mod 6) in blue, even numbers congruent to 2 or 4 (mod 6) in red, and numbers divisible by 6 in amber.
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MAPLE
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ina := proc(n) false end: # adapted from code for A352950
a := proc (n) option remember; local k;
if n < 4 then k := 2*n-1
else for k from 2 while ina(k) or igcd(k, a(n-1)) <> 1 or igcd(k, a(n-2)) <>1
do
end do
end if; ina(k):= true; k
end proc:
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MATHEMATICA
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nn = 66, c[_] = 0; Array[Set[{a[#1], c[#2]}, {#2, #1}] & @@ {#, 2 # - 1} &, 3]; u = 2; Do[k = u; m = LCM @@ Array[a[i - #] &, 2]; While[Nand[c[k] == 0, CoprimeQ[m, k]], k++]; Set[{a[i], c[k]}, {k, i}]; If[a[i] == u, While[c[u] > 0, u++]], {i, 4, nn}]; Array[a, nn]
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PROG
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(Python)
from math import gcd
from itertools import islice
def agen(): # generator of terms
aset, b, c = {1, 3, 5}, 3, 5
yield from [1, b, c]
while True:
k = 1
while k in aset or any(gcd(t, k) != 1 for t in [b, c]): k+= 1
b, c = c, k
aset.add(k)
yield k
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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