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Number of maximal strictly increasing runs in the weakly increasing prime factors of n.
14

%I #6 Aug 05 2024 08:44:07

%S 0,1,1,2,1,1,1,3,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,5,1,1,

%T 1,3,1,1,1,3,1,1,1,2,2,1,1,4,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,6,1,1,1,2,

%U 1,1,1,4,1,1,2,2,1,1,1,4,4,1,1,2,1,1,1

%N Number of maximal strictly increasing runs in the weakly increasing prime factors of n.

%C For n > 1, this is one more than the number of adjacent equal terms in the multiset of prime factors of n.

%H Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.

%F For n > 1, a(n) = A046660(n) + 1 = A001222(n) - A001221(n) + 1.

%e The prime factors of 540 are {2,2,3,3,3,5}, with maximal strictly increasing runs ({2},{2,3},{3},{3,5}), so a(540) = 4.

%t Table[Length[Split[Flatten[ConstantArray@@@FactorInteger[n]],Less]],{n,100}]

%Y For compositions we have A124768, row-lengths of A374683, sum A374684.

%Y For sum of prime indices we have A374706.

%Y Row-lengths of A375128.

%Y A112798 lists prime indices:

%Y - distinct A001221

%Y - length A001222

%Y - leader A055396

%Y - sum A056239

%Y - reverse A296150

%Y Cf. A034296, A046660, A066328, A141199, A141809, A279790, A320324, A333213, A358836, A374700.

%K nonn

%O 1,4

%A _Gus Wiseman_, Aug 04 2024