OFFSET
1,1
COMMENTS
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.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log-log scatterplot of a(n), n = 1..15450, showing records in red, local minima in blue, primes in green, highlighting fixed points in amber.
EXAMPLE
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.
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Scott R. Shannon, Aug 30 2021
STATUS
approved