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%I #13 Sep 02 2020 23:04:37
%S 1,1,1,2,1,1,1,3,2,1,1,3,1,1,1,5,1,3,1,3,1,1,1,5,2,1,3,3,1,1,1,7,1,1,
%T 1,6,1,1,1,5,1,1,1,3,3,1,1,9,2,3,1,3,1,5,1,5,1,1,1,4,1,1,3,11,1,1,1,3,
%U 1,1,1,11,1,1,3,3,1,1,1,9,5,1,1,4,1,1
%N Number of factorizations of n where each factor belongs to A130091 (numbers with distinct prime multiplicities).
%C A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.
%e The a(n) factorizations for n = 2, 4, 8, 60, 16, 36, 32, 48:
%e 2 4 8 5*12 16 4*9 32 48
%e 2*2 2*4 3*20 4*4 3*12 4*8 4*12
%e 2*2*2 3*4*5 2*8 3*3*4 2*16 3*16
%e 2*2*3*5 2*2*4 2*18 2*4*4 3*4*4
%e 2*2*2*2 2*2*9 2*2*8 2*24
%e 2*2*3*3 2*2*2*4 2*3*8
%e 2*2*2*2*2 2*2*12
%e 2*2*3*4
%e 2*2*2*2*3
%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 Table[Length[facsusing[Select[Range[2,n],UnsameQ@@Last/@FactorInteger[#]&],n]],{n,100}]
%Y A327523 is the case when n is restricted to belong to A130091 also.
%Y A001055 counts factorizations.
%Y A007425 counts divisors of divisors.
%Y A045778 counts strict factorizations.
%Y A074206 counts ordered factorizations.
%Y A130091 lists numbers with distinct prime multiplicities.
%Y A181796 counts divisors with distinct prime multiplicities.
%Y A253249 counts nonempty chains of divisors.
%Y A281116 counts factorizations with no common divisor.
%Y A302696 lists numbers whose prime indices are pairwise coprime.
%Y A305149 counts stable factorizations.
%Y A320439 counts factorizations using A289509.
%Y A327498 gives the maximum divisor with distinct prime multiplicities.
%Y A336500 counts divisors of n in A130091 with quotient also in A130091.
%Y A336568 = not a product of two numbers with distinct prime multiplicities.
%Y A336569 counts maximal chains of elements of A130091.
%Y A337256 counts chains of divisors.
%Y Cf. A071625, A080688, A098859, A118914, A124010, A167865, A294068, A303707, A305150, A322453, A336420, A336570, A336571.
%K nonn
%O 1,4
%A _Gus Wiseman_, Aug 03 2020