cp's OEIS Frontend

A336736 Number of factorizations of n whose distinct factors have disjoint prime signatures.

%I A336736 #8 Aug 06 2020 23:28:59
%S A336736 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 A336736 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 A336736 1,1,1,5,1,1,2,2,1,1,1,4,5,1,1,2,1,1,1
%N A336736 Number of factorizations of n whose distinct factors have disjoint prime signatures.
%C A336736 A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.
%e A336736 The a(n) factorizations for n = 36, 360, 720, 192, 288:
%e A336736   (36)     (360)    (720)     (192)      (288)
%e A336736   (6*6)    (5*72)   (8*90)    (3*64)     (8*36)
%e A336736   (2*2*9)  (8*45)   (9*80)    (4*48)     (9*32)
%e A336736   (3*3*4)  (9*40)   (10*72)   (6*32)     (16*18)
%e A336736            (10*36)  (16*45)   (12*16)    (2*144)
%e A336736            (5*8*9)  (5*144)   (3*8*8)    (6*6*8)
%e A336736                     (5*9*16)  (4*6*8)    (2*2*72)
%e A336736                     (8*9*10)  (3*4*16)   (2*9*16)
%e A336736                               (3*4*4*4)  (3*3*32)
%e A336736                                          (2*2*8*9)
%e A336736                                          (3*3*4*8)
%e A336736                                          (2*2*2*36)
%e A336736                                          (2*2*2*2*2*9)
%t A336736 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A336736 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A336736 prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];
%t A336736 Table[Length[Select[facs[n],stableQ[#,Intersection[prisig[#1],prisig[#2]]!={}&]&]],{n,100}]
%Y A336736 A001055 counts factorizations.
%Y A336736 A118914 is sorted prime signature.
%Y A336736 A124010 is prime signature.
%Y A336736 A336737 counts factorizations with intersecting signatures.
%Y A336736 Cf. A000372, A003182, A006126, A109298, A112798, A293606, A294068, A305844, A321469, A336424.
%K A336736 nonn
%O A336736 1,4
%A A336736 _Gus Wiseman_, Aug 06 2020