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Number of factorizations of the n-th uniform number A072774(n) into uniform numbers > 1.
2

%I #6 Sep 18 2019 04:57:08

%S 1,1,1,2,1,2,1,3,2,2,1,1,2,2,5,1,1,2,2,1,2,2,3,1,5,1,7,2,2,2,7,1,2,2,

%T 1,5,1,2,1,2,2,1,2,2,2,1,1,2,11,2,5,1,2,5,1,1,2,2,5,1,5,2,1,2,2,2,1,2,

%U 2,2,2,1,7,1,5,1,5,2,1,1,5,2,1,5,2,2,2

%N Number of factorizations of the n-th uniform number A072774(n) into uniform numbers > 1.

%C A number is uniform if its prime multiplicities are all equal, meaning it is a power of a squarefree number. Uniform numbers are listed in A072774.

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vSX9dPMGJhxB8rOknCGvOs6PiyhupdWNpqLsnphdgU6MEVqFBnWugAXidDhwHeKqZe_YnUqYeGOXsOk/pub">Sequences counting and encoding certain classes of multisets</a>

%e The a(31) = 7 factorizations of 36 into uniform numbers together with the corresponding multiset partitions of {1,1,2,2}:

%e (2*2*3*3) {{1},{1},{2},{2}}

%e (2*2*9) {{1},{1},{2,2}}

%e (2*3*6) {{1},{2},{1,2}}

%e (3*3*4) {{2},{2},{1,1}}

%e (4*9) {{1,1},{2,2}}

%e (6*6) {{1,2},{1,2}}

%e (36) {{1,1,2,2}}

%t nn=100;

%t facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];

%t y=Select[Range[nn],SameQ@@Last/@FactorInteger[#]&];

%t Table[Length[facsusing[Rest[y],n]],{n,y}];

%Y See link for additional cross-references.

%Y Cf. A000961, A001055, A005117, A007947, A071625, A112798.

%K nonn

%O 1,4

%A _Gus Wiseman_, Sep 17 2019