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Numbers that are a smallest number with k pairs of successive divisors, for some k.
4

%I #7 Oct 18 2019 16:45:20

%S 1,2,6,12,60,72,180,360,420,840,1260,2520,3780,5040,13860,27720,36960,

%T 41580,55440,83160,166320,277200,360360,471240,491400,720720,1081080,

%U 1113840,2162160,2827440,3341520,4324320,5405400,6126120

%N Numbers that are a smallest number with k pairs of successive divisors, for some k.

%C A sorted version of A287142.

%e The divisors of 72 are {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72}, with pairs of successive divisors {{1, 2}, {2, 3}, {3, 4}, {8, 9}}, and no smaller number has 4 successive pairs, so 72 belongs to the sequence.

%t dat=Table[Count[Differences[Divisors[n]],1],{n,10000}];

%t Sort[Table[Position[dat,i][[1,1]],{i,Union[dat]}]]

%Y Sorted positions of first appearances in A129308.

%Y The longest run of divisors of n has length A055874(n).

%Y Numbers whose divisors > 1 have no non-singleton runs are A088725.

%Y The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).

%Y The smallest number whose divisors have a longest run of length n is A328449(n).

%Y Cf. A000005, A003601, A027750, A033676, A060680, A060681, A072627, A181063, A199970, A287142, A328165.

%K nonn

%O 0,2

%A _Gus Wiseman_, Oct 15 2019