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A351626
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a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that shares a factor with both the largest and second largest value of all previous terms.
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5
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1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 15, 22, 24, 26, 28, 30, 21, 32, 34, 36, 38, 40, 42, 35, 44, 33, 46, 48, 50, 45, 52, 54, 39, 56, 58, 60, 62, 64, 66, 68, 51, 70, 72, 63, 74, 76, 78, 57, 80, 65, 75, 82, 84, 86, 88, 90, 55, 92, 69, 94, 96, 98, 100, 102, 85, 104, 106, 108, 110, 99, 105, 112, 77
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OFFSET
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1,2
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COMMENTS
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The sequence contains no primes or prime powers other than the powers of 2. As the sequence starts with 2 and 4, these terms being the largest and second largest values, the following terms will be even. This pattern continues until a term equal to the product of two or more odd primes occurs that shares a factor with the previous two largest even values and is smaller than the largest value plus 2. It is not possible for this value to be a prime larger than 2, or a power of such a prime, as the two terms with which it must share a factor differ by 2. It therefore cannot be between them either so it must be less than the second largest even term. Thus the next term after this odd composite must still share a factor with the two largest even values, and this will be the largest value plus 2 or another smaller odd composite. Therefore two more even values eventually become the two largest terms again, and thus the pattern of the two largest even terms, differing by two, interrupted by odd composites continues. Therefore no primes or prime powers other than powers of 2 will occur.
In the first 200000 terms the maximum run of even and odd terms is twelve and seven respectively; it is unknown if these runs have a maximum number of terms or are unbounded. The fixed points beyond 2 in the same range are 573, 597, 633, 6487, 21865, 22115, although it is likely more exist.
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LINKS
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EXAMPLE
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a(5) = 8 as the largest and second-largest values of all previous terms are a(4) = 6 and a(3) = 4, and 8 is the smallest unused number that shares a factor with both of these values.
a(12) = 15 as the largest and second-largest values of all previous terms are a(11) = 20 and a(10) = 18, and 15 is the smallest unused number that shares a factor with both of these values.
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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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