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Numbers whose divisors do not have weakly decreasing run-lengths.
5

%I #5 Oct 09 2019 10:03:01

%S 56,72,110,112,132,144,156,182,210,216,224,240,264,272,288,306,312,

%T 342,364,380,392,396,420,432,440,448,462,468,480,506,528,544,550,552,

%U 576,600,612,616,624,648,650,684,702,720,728,756,760,770,780,784,792,812

%N Numbers whose divisors do not have weakly decreasing run-lengths.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Run_(cards)">Run (cards)</a>

%e The divisors of 56 are {1, 2, 4, 7, 8, 14, 28, 56}, with runs {{1, 2}, {4}, {7, 8}, {14}, {28}, {56}}, with lengths (2, 1, 2, 1, 1, 1), which are not weakly decreasing, so 56 is in the sequence.

%t Select[Range[1000],!GreaterEqual@@Length/@Split[Divisors[#],#2==#1+1&]&]

%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 number of successive pairs of divisors of n is A129308(n).

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

%Y Cf. A000005, A027750, A033676, A060680, A060775, A119313, A181063, A199970.

%K nonn

%O 1,1

%A _Gus Wiseman_, Oct 07 2019