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%I #8 Aug 06 2020 23:28:59
%S 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,5,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,7,1,1,
%T 1,4,1,1,1,3,1,1,1,2,2,1,1,4,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,11,1,1,1,2,
%U 1,1,1,5,1,1,2,2,1,1,1,4,5,1,1,2,1,1,1
%N Number of factorizations of n whose distinct factors have disjoint prime signatures.
%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 = 36, 360, 720, 192, 288:
%e (36) (360) (720) (192) (288)
%e (6*6) (5*72) (8*90) (3*64) (8*36)
%e (2*2*9) (8*45) (9*80) (4*48) (9*32)
%e (3*3*4) (9*40) (10*72) (6*32) (16*18)
%e (10*36) (16*45) (12*16) (2*144)
%e (5*8*9) (5*144) (3*8*8) (6*6*8)
%e (5*9*16) (4*6*8) (2*2*72)
%e (8*9*10) (3*4*16) (2*9*16)
%e (3*4*4*4) (3*3*32)
%e (2*2*8*9)
%e (3*3*4*8)
%e (2*2*2*36)
%e (2*2*2*2*2*9)
%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 A336737 counts factorizations with intersecting signatures.
%Y Cf. A000372, A003182, A006126, A109298, A112798, A293606, A294068, A305844, A321469, A336424.
%K nonn
%O 1,4
%A _Gus Wiseman_, Aug 06 2020
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