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A319118 Number of multimin tree-factorizations of n.

%I A319118 #10 Dec 14 2024 07:29:13
%S A319118 1,1,1,2,1,2,1,6,2,2,1,8,1,2,2,24,1,6,1,8,2,2,1,42,2,2,6,8,1,8,1,112,
%T A319118 2,2,2,38,1,2,2,42,1,8,1,8,8,2,1,244,2,6,2,8,1,24,2,42,2,2,1,58,1,2,8,
%U A319118 568,2,8,1,8,2,8,1,268,1,2,6,8,2,8,1,244,24
%N A319118 Number of multimin tree-factorizations of n.
%C A319118 A multimin factorization of n is an ordered factorization of n into factors greater than 1 such that the sequence of minimal primes dividing each factor is weakly increasing. A multimin tree-factorization of n is either the number n itself or a sequence of multimin tree-factorizations, one of each factor in a multimin factorization of n with at least two factors.
%F A319118 a(prime^n) = A118376(n).
%F A319118 a(product of n distinct primes) = A005804(n).
%e A319118 The a(12) = 8 multimin tree-factorizations:
%e A319118   12,
%e A319118   (2*6), (4*3), (6*2), (2*2*3),
%e A319118   (2*(2*3)), ((2*2)*3), ((2*3)*2).
%e A319118 Or as series-reduced plane trees of multisets:
%e A319118   112,
%e A319118   (1,12), (11,2), (12,1), (1,1,2),
%e A319118   (1,(1,2)), ((1,1),2), ((1,2),1).
%t A319118 facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
%t A319118 mmftrees[n_]:=Prepend[Join@@(Tuples[mmftrees/@#]&/@Select[Join@@Permutations/@Select[facs[n],Length[#]>1&],OrderedQ[FactorInteger[#][[1,1]]&/@#]&]),n];
%t A319118 Table[Length[mmftrees[n]],{n,100}]
%Y A319118 Cf. A001055, A005804, A020639, A118376, A196545, A255397, A281113, A281118, A281119, A295279, A317545, A317546, A318577, A319119.
%K A319118 nonn
%O A319118 1,4
%A A319118 _Gus Wiseman_, Sep 10 2018