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A327400 Number of factorizations of n that are constant or whose factors are relatively prime.

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%I A327400 #4 Sep 23 2019 13:41:43
%S A327400 1,1,1,2,1,2,1,2,2,2,1,3,1,2,2,3,1,3,1,3,2,2,1,4,2,2,2,3,1,5,1,2,2,2,
%T A327400 2,7,1,2,2,4,1,5,1,3,3,2,1,6,2,3,2,3,1,4,2,4,2,2,1,9,1,2,3,4,2,5,1,3,
%U A327400 2,5,1,10,1,2,3,3,2,5,1,6,3,2,1,9,2,2,2
%N A327400 Number of factorizations of n that are constant or whose factors are relatively prime.
%C A327400 First differs from A327399 at a(24) = 4, A327399(24) = 3.
%F A327400 a(n) = A281116(n) + A089723(n).
%e A327400 The factorizations of 2, 4, 12, 24, 30, 36, 48, and 60 that are constant or whose factors are relatively prime:
%e A327400   2   4     12      24        30      36        48          60
%e A327400       2*2   3*4     3*8       5*6     4*9       3*16        3*20
%e A327400             2*2*3   2*3*4     2*15    6*6       2*3*8       4*15
%e A327400                     2*2*2*3   3*10    2*2*9     3*4*4       5*12
%e A327400                               2*3*5   2*3*6     2*2*3*4     2*5*6
%e A327400                                       3*3*4     2*2*2*2*3   3*4*5
%e A327400                                       2*2*3*3               2*2*15
%e A327400                                                             2*3*10
%e A327400                                                             2*2*3*5
%t A327400 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A327400 Table[Length[Select[facs[n],#=={}||Length[Union[#]]==1||GCD@@#==1&]],{n,100}]
%Y A327400 Constant factorizations are A089723.
%Y A327400 Cf. A007359, A051424, A281116, A302569, A304709, A304711, A319269, A327399.
%K A327400 nonn
%O A327400 1,4
%A A327400 _Gus Wiseman_, Sep 22 2019