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Number of complete multimin tree-factorizations of Heinz numbers of integer partitions of n.
1

%I #8 Sep 14 2018 11:18:53

%S 1,2,5,18,74,344,1679,8548,44690,238691,1295990,7132509

%N Number of complete multimin tree-factorizations of Heinz numbers of integer partitions of n.

%C 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 at least two multimin tree-factorizations, one of each factor in a multimin factorization of n. A multimin tree-factorization is complete if the leaves are all prime numbers.

%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

%e The a(3) = 5 trees are: 5, (2*3), (2*2*2), (2*(2*2)), ((2*2)*2).

%e The a(4) = 18 trees (normalized with prime(n) -> n):

%e 4,

%e (13), (22), (112), (1111),

%e (1(12)), ((12)1), ((11)2),

%e (11(11)), (1(11)1), ((11)11), (1(111)), ((111)1), ((11)(11)),

%e (1(1(11))), (1((11)1)), ((1(11))1), (((11)1)1).

%t facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];

%t mmftrees[n_]:=Prepend[Join@@(Tuples[mmftrees/@#]&/@Select[Join@@Permutations/@Select[facs[n],Length[#]>1&],OrderedQ[FactorInteger[#][[1,1]]&/@#]&]),n];

%t Table[Sum[Length[Select[mmftrees[k],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{k,Times@@Prime/@#&/@IntegerPartitions[n]}],{n,10}]

%Y Cf. A000311, A001003, A001055, A020639, A255397, A281113, A281118, A281119, A295281, A317545, A317546, A318577, A319118, A319119.

%K nonn,more

%O 1,2

%A _Gus Wiseman_, Sep 11 2018

%E a(11)-a(12) from _Robert Price_, Sep 14 2018