A319002 - OEIS

%I #8 Sep 11 2018 17:01:24

%S 1,1,1,2,1,2,1,4,2,2,1,5,1,2,2,8,1,4,1,5,2,2,1,12,2,2,4,5,1,6,1,16,2,

%T 2,2,11,1,2,2,12,1,5,1,5,4,2,1,28,2,4,2,5,1,8,2,12,2,2,1,18,1,2,5,32,

%U 2,6,1,5,2,6,1,29,1,2,4,5,2,5,1,28,8,2,1

%N Number of ordered factorizations of n where the sequence of GCDs of prime indices (A289508) of the factors is weakly increasing.

%C Also the number of ordered multiset partitions of the multiset of prime indices of n where the sequence of GCDs of the blocks is weakly increasing. If we form a multiorder by treating integer partitions (a,...,z) as multiarrows GCD(a,...,z) <= {z,...,a}, then a(n) is the number of triangles whose composite ground is the integer partition with Heinz number n.

%e The a(36) = 11 ordered factorizations:

%e   (2*2*3*3),

%e   (2*2*9), (2*6*3), (6*2*3), (4*3*3),

%e   (2*18), (18*2), (12*3), (4*9), (6*6),

%e   (36).

%e The a(36) = 11 ordered multiset partitions:

%e      {{1,1,2,2}}

%e     {{1},{1,2,2}}

%e     {{1,2,2},{1}}

%e     {{1,1,2},{2}}

%e     {{1,1},{2,2}}

%e     {{1,2},{1,2}}

%e    {{1},{1},{2,2}}

%e    {{1},{1,2},{2}}

%e    {{1,2},{1},{2}}

%e    {{1,1},{2},{2}}

%e   {{1},{1},{2},{2}}

%t facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];

%t gix[n_]:=GCD@@PrimePi/@If[n==1,{},FactorInteger[n]][[All,1]];

%t Table[Length[Select[Join@@Permutations/@facs[n],OrderedQ[gix/@#]&]],{n,100}]

%Y Cf. A000837, A007716, A001970, A056239, A063834, A289508, A296150, A316223, A317545, A318559, A319001, A319004.

%K nonn

%O 1,4

%A _Gus Wiseman_, Sep 07 2018