cp's OEIS Frontend

A319119 Number of multimin tree-factorizations of Heinz numbers of integer partitions of n.

%I A319119 #9 Sep 14 2018 10:56:19
%S A319119 1,3,9,37,173,921,5185,30497,184469,1140413,7170085,45704821
%N A319119 Number of multimin tree-factorizations of Heinz numbers of integer partitions of n.
%C A319119 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.
%C A319119 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%e A319119 The a(3) = 9 multimin tree-factorizations:
%e A319119   5, 6, 8,
%e A319119   (2*3), (2*4), (4*2), (2*2*2),
%e A319119   (2*(2*2)), ((2*2)*2).
%e A319119 Or as series-reduced plane trees of multisets:
%e A319119   3, 12, 111,
%e A319119   (1,2), (1,11), (11,1), (1,1,1),
%e A319119   (1,(1,1)), ((1,1),1).
%t A319119 facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
%t A319119 mmftrees[n_]:=Prepend[Join@@(Tuples[mmftrees/@#]&/@Select[Join@@Permutations/@Select[facs[n],Length[#]>1&],OrderedQ[FactorInteger[#][[1,1]]&/@#]&]),n];
%t A319119 Table[Sum[Length[mmftrees[k]],{k,Times@@Prime/@#&/@IntegerPartitions[n]}],{n,7}]
%Y A319119 Cf. A001055, A020639, A196545, A255397, A281113, A281118, A281119, A295279, A317545, A317546, A319118.
%K A319119 nonn,more
%O A319119 1,2
%A A319119 _Gus Wiseman_, Sep 10 2018
%E A319119 a(11)-a(12) from _Robert Price_, Sep 14 2018