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A251413
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a(n) = 2n-1 if n <= 3, otherwise the smallest odd number not occurring earlier having at least one common factor with a(n-2), but none with a(n-1).
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3
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1, 3, 5, 9, 25, 21, 55, 7, 11, 35, 33, 49, 15, 77, 27, 91, 45, 13, 51, 65, 17, 39, 85, 57, 115, 19, 23, 95, 69, 125, 63, 145, 81, 29, 75, 203, 93, 119, 31, 105, 341, 87, 121, 111, 143, 37, 99, 185, 117, 155, 123, 175, 41, 133
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OFFSET
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1,2
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COMMENTS
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Conjecture 1: Sequence is a permutation of the odd numbers.
Conjecture 2: The odd primes occur in the sequence in their natural order.
The following properties are known (the proofs are analogous to the proofs for the corresponding facts about A098550).
1. The sequence is infinite.
2. At least one-third of the terms are composite.
3. For any odd prime p, there is a term that is divisible by p.
4. Let a(n_p) be the first term that is divisible by p. Then a(n_p) = q*p where q is an odd prime less than p. If p < r are primes then n_p < n_r.
(End)
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REFERENCES
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L. Edson Jeffery, Posting to Sequence Fans Mailing List, Dec 01 2014
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LINKS
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MAPLE
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N:= 10^3: # to get a(1) to a(n) where a(n+1) is the first term > N
B:= Vector(N, datatype=integer[4]):
for n from 1 to 3 do A[n]:= 2*n-1: od:
for n from 4 do
for k from 4 to N do
if B[k] = 0 and igcd(2*k-1, A[n-1]) = 1 and igcd(2*k-1, A[n-2]) > 1 then
A[n]:= 2*k-1;
B[k]:= 1;
break
fi
od:
if 2*k-1 > N then break fi
od:
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MATHEMATICA
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max = 54; f = True; a = {1, 3, 5}; NN = Range[4, 1000]; s = 2*NN - 1; While[TrueQ[f], For[k = 1, k <= Length[s], k++, If[Length[a] < max, If[GCD[a[[-1]], s[[k]]] == 1 && GCD[a[[-2]], s[[k]]] > 1, a = Append[a, s[[k]]]; s = Delete[s, k]; k = 0; Break], f = False]]]; a (* L. Edson Jeffery, Dec 02 2014 *)
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PROG
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(Python)
from fractions import gcd
A251413_list, l1, l2, s, b = [1, 3, 5], 5, 3, 7, {}
for _ in range(1, 10**4):
....i = s
....while True:
........if not i in b and gcd(i, l1) == 1 and gcd(i, l2) > 1:
............l2, l1, b[i] = l1, i, True
............while s in b:
................b.pop(s)
................s += 2
............break
(Haskell)
import Data.List (delete)
a251413 n = a251413_list !! (n-1)
a251413_list = 1 : 3 : 5 : f 3 5 [7, 9 ..] where
f u v ws = g ws where
g (x:xs) = if gcd x u > 1 && gcd x v == 1
then x : f v x (delete x ws) else g xs
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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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