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A316979 Number of strict factorizations of n into factors > 1 with no equivalent primes. 7

%I

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

%T 1,4,1,1,1,5,1,1,1,3,3,1,1,7,1,3,1,3,1,5,1,5,1,1,1,6,1,1,3,4,1,1,1,3,

%U 1,1,1,9,1,1,3,3,1,1,1,7,2,1,1,6,1,1,1

%N Number of strict factorizations of n into factors > 1 with no equivalent primes.

%C In a factorization, two primes are equivalent if each factor has in its prime factorization the same multiplicity of both primes. For example, in 60 = (2*30) the primes {3, 5} are equivalent but {2, 3} and {2, 5} are not.

%F a(prime^n) = A000009(n).

%e The a(24) = 5 factorizations are (2*3*4), (2*12), (3*8), (4*6), (24).

%e The a(36) = 4 factorizations are (2*3*6), (2*18), (3*12), (4*9).

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

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

%t dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];

%t Table[Length[Select[facs[n],And[UnsameQ@@#,UnsameQ@@dual[primeMS/@#]]&]],{n,100}]

%Y Cf. A000009, A001055, A007716, A007717, A020555, A045778, A130091, A162247, A281116.

%Y Cf. A316974, A316978, A316980, A316981.

%K nonn

%O 1,8

%A _Gus Wiseman_, Jul 18 2018

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Last modified July 15 03:31 EDT 2020. Contains 335762 sequences. (Running on oeis4.)