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%I #6 Aug 06 2020 23:29:08
%S 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 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 2,5,1,9,1,2,3,3,2,5,1,6,3,2,1,9,2,2,2
%N Number of factorizations of n whose factors have pairwise intersecting prime signatures.
%C First differs from A327400 at a(72) = 9, A327400(72) = 10.
%C A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.
%e The a(n) factorizations for n = 2, 4, 12, 24, 30, 36, 60:
%e (2) (4) (12) (24) (30) (36) (60)
%e (2*2) (2*6) (2*12) (5*6) (4*9) (2*30)
%e (2*2*3) (2*2*6) (2*15) (6*6) (3*20)
%e (2*2*2*3) (3*10) (2*18) (5*12)
%e (2*3*5) (3*12) (6*10)
%e (2*3*6) (2*5*6)
%e (2*2*3*3) (2*2*15)
%e (2*3*10)
%e (2*2*3*5)
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];
%t Table[Length[Select[facs[n],stableQ[#,Intersection[prisig[#1],prisig[#2]]=={}&]&]],{n,100}]
%Y A001055 counts factorizations.
%Y A118914 is sorted prime signature.
%Y A124010 is prime signature.
%Y A336736 counts factorizations with disjoint signatures.
%Y Cf. A003182, A051185, A305843, A305844, A305854, A306006, A319752, A319787, A319789, A321469, A336424.
%K nonn
%O 1,4
%A _Gus Wiseman_, Aug 06 2020
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