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A347404
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a(1) = 3; for n > 2, a(n) is the smallest distinct positive integer such that gcd(a(n), a(n-k)) > 1 for each divisor k of a(n) such that k < n.
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1
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3, 6, 9, 12, 15, 18, 10, 2, 14, 4, 22, 8, 24, 16, 20, 28, 26, 30, 34, 32, 36, 40, 5, 50, 60, 38, 42, 44, 46, 48, 54, 52, 58, 56, 62, 64, 66, 68, 70, 72, 74, 78, 39, 90, 57, 21, 51, 69, 84, 87, 33, 27, 96, 81, 63, 7, 126, 45, 168, 99, 93, 75, 95, 19, 114, 105, 77, 198, 153, 165, 11, 132, 55, 595
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OFFSET
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1,1
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COMMENTS
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The majority of terms are concentrated along a line whose slope is approximately 1.3. Occasionally though there are terms which correspond to the smallest unused number up to that point, and these tend to lead to a subsequent very large term. For example a(499) = 628, a(500) = 682, a(501) = 31, a(502) = 14322. Other large terms appear seemingly at random, for example a(15449) = 19880, a(15450) = 19099, a(15451) = 74962230.
It is likely all numbers > 1 eventually appear. The smallest number not seen after 20000 terms is 89.
Note that if the sequence starts with 2 then the terms are just all the increasing even numbers.
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LINKS
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EXAMPLE
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a(2) = 6. As a(1) = 3 the next term must be a multiple of 3, and the smallest unused such number is 6. Note that as a(2-2), a(2-3) and a(2-6) are not defined these are ignored.
a(7) = 10. As a(6) = 18 the next term must have 2 and/or 3 as divisors. If it has 2 as a divisor is must also have 3 and/or 5 as a divisor as a(8-2) = a(6) = 15. The smallest unused number satisfying these is 10. Note that as 5 is a divisor of 10 it must be that a(7-5) = a(2) = 6 has 2 or 5 as a divisor, which is true.
a(8) = 2. As a(7) = 10 the next term must have 2 and/or 5 as a divisor. As a(6) = 18 also has 2 as a divisor a(8) = 2 is the next smallest unused term.
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MATHEMATICA
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nn = 74; c[_] = 0; a[1] = 3; c[3] = 1; u = 2; While[c[u] > 0, u++]; Do[k = u; While[Nand[c[k] == 0, AllTrue[TakeWhile[Divisors[k], # < i &], ! CoprimeQ[a[i - #], k] &]], k++]; Set[{a[i], c[k]}, {k, i}], {i, Length[s] + 1, nn}]; Array[a, nn] (* Michael De Vlieger, Apr 12 2022 *)
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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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