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A336737
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Number of factorizations of n whose factors have pairwise intersecting prime signatures.
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5
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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, 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, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 6, 3, 2, 1, 9, 2, 2, 2
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OFFSET
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1,4
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COMMENTS
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A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.
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LINKS
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EXAMPLE
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The a(n) factorizations for n = 2, 4, 12, 24, 30, 36, 60:
(2) (4) (12) (24) (30) (36) (60)
(2*2) (2*6) (2*12) (5*6) (4*9) (2*30)
(2*2*3) (2*2*6) (2*15) (6*6) (3*20)
(2*2*2*3) (3*10) (2*18) (5*12)
(2*3*5) (3*12) (6*10)
(2*3*6) (2*5*6)
(2*2*3*3) (2*2*15)
(2*3*10)
(2*2*3*5)
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MATHEMATICA
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facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
prisig[n_]:=If[n==1, {}, Last/@FactorInteger[n]];
Table[Length[Select[facs[n], stableQ[#, Intersection[prisig[#1], prisig[#2]]=={}&]&]], {n, 100}]
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CROSSREFS
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A336736 counts factorizations with disjoint signatures.
Cf. A003182, A051185, A305843, A305844, A305854, A306006, A319752, A319787, A319789, A321469, A336424.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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