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A270139
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a(n)=n when n<=3, otherwise a(n) is the smallest unused positive integer which is not coprime to the two previous terms.
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8
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1, 2, 3, 6, 9, 12, 15, 10, 5, 20, 25, 30, 35, 14, 7, 21, 28, 18, 4, 8, 16, 22, 24, 26, 32, 34, 36, 38, 40, 42, 44, 33, 11, 55, 66, 45, 27, 39, 48, 51, 54, 57, 60, 63, 56, 49, 70, 77, 84, 88, 46, 50, 52, 58, 62, 64, 68, 72, 74, 76, 78, 80, 65, 75, 85, 90, 95, 100, 105, 96
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OFFSET
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1,2
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COMMENTS
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Other possible conditions on a(n) with respect to its common factors with a(n-2) and a(n-1) lead to the following:
Coprime to the latter and not the former: A098550.
Coprime to the former and not the latter: with any initial conditions, the sequence "paints itself into a corner", i.e., is finite. With the added condition of a(n) having an extra prime factor not contained in a(n-1), it is A336957.
Coprime to the latter, regardless of the former: simply A000027.
Coprime to the former, regardless of the latter: A121216.
Non-coprime to the latter, regardless of the former: A064413.
Non-coprime to the former, regardless of the latter: A121217.
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LINKS
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EXAMPLE
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a(12) = 30, a(13) = 35, so a(14) must have common factors (possibly different) with 30 and 35, and the smallest unused number with that property turns out to be 14, so a(14) = 14.
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MATHEMATICA
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a = {1, 2, 3}; Do[k = 1; While[(MemberQ[a, k] || GCD[a[[-1]], k] == 1 || GCD[a[[-2]], k] == 1), k++]; AppendTo[a, k], {n, 2, 68}]; a
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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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