Hello MathFun & SeqFans, Rule: < Add to n its second smallest non-divider. Loop. > Let's start with n = 7, for instance Is 1 a divider of 7? yes 2 no 3 no --> then new n = 7+3 = 10 Is 1 a divider of 10? yes 2 yes 3 no 4 no --> then new n =10+4= 14 Is 1 a divider of 14? yes 2 yes 3 no 4 no --> then new n =14+4= 18 Is 1 a divider of 18? yes 2 yes 3 yes 4 no 5 no --> then new n =18+5= 23 ... etc. Sequence starting with 7 is: 7,10,14,18,23,... Sequence starting with 1 is: 1,4,9,13,16,21,... [not in the OEIS] Sequence starting with 2 is: 2,6,11,14,18,23,... [merges with "7-seq"] Sequence starting with 3 is: 3,7,10,14,... [merges with "7-seq"] Sequence starting with 5 is: 5,8,13,16,... [merges with "1-seq"] etc. We might map those sequences like this: 1--4--9---13--16--21--25--28--32--37--40 ... | | | 5--8---+ 29--+ | | 2--6--11--14--18--23--26--30------+ | | | 3--7--10--+ | 33--+ | 12--19--22--+ | | 15--+ | | 17--20--+ 24--31--34--38--42 ... | | 27--+ 35--+ What number starts the longest sequence containing 2008? Best, E. --- P.-S. This rule is not very productive: < Add to n its first smallest non-divider. Loop. > What about: < Add to n its third smallest non-divider. Loop. > Etc.