cp's OEIS Frontend

A319121 Number of complete multimin tree-factorizations of Heinz numbers of integer partitions of n.

%I A319121 #8 Sep 14 2018 11:18:53
%S A319121 1,2,5,18,74,344,1679,8548,44690,238691,1295990,7132509
%N A319121 Number of complete multimin tree-factorizations of Heinz numbers of integer partitions of n.
%C A319121 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 A319121 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%e A319121 The a(3) = 5 trees are: 5, (2*3), (2*2*2), (2*(2*2)), ((2*2)*2).
%e A319121 The a(4) = 18 trees (normalized with prime(n) -> n):
%e A319121   4,
%e A319121   (13), (22), (112), (1111),
%e A319121   (1(12)), ((12)1), ((11)2),
%e A319121   (11(11)), (1(11)1), ((11)11), (1(111)), ((111)1), ((11)(11)),
%e A319121   (1(1(11))), (1((11)1)), ((1(11))1), (((11)1)1).
%t A319121 facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
%t A319121 mmftrees[n_]:=Prepend[Join@@(Tuples[mmftrees/@#]&/@Select[Join@@Permutations/@Select[facs[n],Length[#]>1&],OrderedQ[FactorInteger[#][[1,1]]&/@#]&]),n];
%t A319121 Table[Sum[Length[Select[mmftrees[k],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{k,Times@@Prime/@#&/@IntegerPartitions[n]}],{n,10}]
%Y A319121 Cf. A000311, A001003, A001055, A020639, A255397, A281113, A281118, A281119, A295281, A317545, A317546, A318577, A319118, A319119.
%K A319121 nonn,more
%O A319121 1,2
%A A319121 _Gus Wiseman_, Sep 11 2018
%E A319121 a(11)-a(12) from _Robert Price_, Sep 14 2018