A317545 -id:A317545 - cp's OEIS frontend

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

1, 3, 9, 37, 173, 921, 5185, 30497, 184469, 1140413, 7170085, 45704821

Offset: 1

Gus Wiseman, Sep 10 2018

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.

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

			The a(3) = 9 multimin tree-factorizations:
  5, 6, 8,
  (2*3), (2*4), (4*2), (2*2*2),
  (2*(2*2)), ((2*2)*2).
Or as series-reduced plane trees of multisets:
  3, 12, 111,
  (1,2), (1,11), (11,1), (1,1,1),
  (1,(1,1)), ((1,1),1).

Cf. A001055, A020639, A196545, A255397, A281113, A281118, A281119, A295279, A317545, A317546, A319118.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
mmftrees[n_]:=Prepend[Join@@(Tuples[mmftrees/@#]&/@Select[Join@@Permutations/@Select[facs[n],Length[#]>1&],OrderedQ[FactorInteger[#][[1,1]]&/@#]&]),n];
Table[Sum[Length[mmftrees[k]],{k,Times@@Prime/@#&/@IntegerPartitions[n]}],{n,7}]

a(11)-a(12) from Robert Price, Sep 14 2018

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

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 11 2018

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.

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

			The a(3) = 5 trees are: 5, (2*3), (2*2*2), (2*(2*2)), ((2*2)*2).
The a(4) = 18 trees (normalized with prime(n) -> n):
  4,
  (13), (22), (112), (1111),
  (1(12)), ((12)1), ((11)2),
  (11(11)), (1(11)1), ((11)11), (1(111)), ((111)1), ((11)(11)),
  (1(1(11))), (1((11)1)), ((1(11))1), (((11)1)1).

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

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
mmftrees[n_]:=Prepend[Join@@(Tuples[mmftrees/@#]&/@Select[Join@@Permutations/@Select[facs[n],Length[#]>1&],OrderedQ[FactorInteger[#][[1,1]]&/@#]&]),n];
Table[Sum[Length[Select[mmftrees[k],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{k,Times@@Prime/@#&/@IntegerPartitions[n]}],{n,10}]

a(11)-a(12) from Robert Price, Sep 14 2018

cp's OEIS Frontend

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

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