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Number of factorizations of the n-th number that is 1 or whose prime indices are relatively prime A289509(n - 1) into numbers > 1 satisfying the same conditions.
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%I #6 Sep 18 2019 04:57:42

%S 1,1,2,1,3,1,2,1,1,5,1,2,1,4,1,2,2,7,1,1,1,3,1,4,1,2,1,1,7,1,1,2,1,1,

%T 4,1,5,1,11,2,2,1,2,6,1,1,2,1,1,7,1,3,1,1,4,3,2,1,1,1,12,1,1,3,2,4,1,

%U 1,3,2,7,1,2,1,1,10,1,1,2,1,15,1,5,1,1,4

%N Number of factorizations of the n-th number that is 1 or whose prime indices are relatively prime A289509(n - 1) into numbers > 1 satisfying the same conditions.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. Numbers whose prime indices are relatively prime are A289509.

%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(76) = 10 factorizations of 120 into elements of A289509 together with the corresponding multiset partitions of {1,1,1,2,3}:

%e (2*2*2*15) {{1},{1},{1},{2,3}}

%e (2*2*30) {{1},{1},{1,2,3}}

%e (2*4*15) {{1},{1,1},{2,3}}

%e (2*6*10) {{1},{1,2},{1,3}}

%e (2*60) {{1},{1,1,2,3}}

%e (4*30) {{1,1},{1,2,3}}

%e (6*20) {{1,2},{1,1,3}}

%e (8*15) {{1,1,1},{2,3}}

%e (10*12) {{1,3},{1,1,2}}

%e (120) {{1,1,1,2,3}}

%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],#==1||GCD@@PrimePi/@First/@FactorInteger[#]==1&];

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

%Y See link for additional cross-references.

%Y Cf. A001055, A006530, A112798, A281116, A289509, A318721.

%K nonn

%O 1,3

%A _Gus Wiseman_, Sep 17 2019