A317546 -id:A317546 - cp's OEIS frontend

A319004 Number of ordered factorizations of n where the sequence of LCMs of the prime indices (A290103) of each factor is weakly increasing.

1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 4, 1, 2, 2, 8, 1, 5, 1, 4, 2, 2, 1, 8, 2, 2, 4, 4, 1, 5, 1, 16, 2, 2, 2, 11, 1, 2, 2, 8, 1, 5, 1, 4, 4, 2, 1, 16, 2, 5, 2, 4, 1, 12, 2, 8, 2, 2, 1, 11, 1, 2, 4, 32, 2, 5, 1, 4, 2, 5, 1, 23, 1, 2, 4, 4, 2, 5, 1, 16, 8, 2, 1, 11, 2, 2, 2, 8, 1, 12, 2, 4, 2, 2, 2, 32, 1, 5, 4, 11, 1, 5, 1, 8, 5

Offset: 1

Gus Wiseman, Sep 07 2018

nonn

Also the number of ordered multiset partitions of the multiset of prime indices of n where the sequence of LCMs of the parts is weakly increasing. If we form a multiorder by treating integer partitions (a,...,z) as multiarrows LCM(a,...,z) <= {z,...,a}, then a(n) is the number of triangles whose composite ground is the integer partition with Heinz number n.

			The a(60) = 11 ordered factorizations:
  (2*2*3*5),
  (2*2*15), (2*3*10), (2*6*5), (4*3*5),
  (2*30), (3*20), (4*15), (12*5), (6*10),
  (60).
The a(60) = 11 ordered multiset partitions:
     {{1,1,2,3}}
    {{1},{1,2,3}}
    {{2},{1,1,3}}
    {{1,1,2},{3}}
    {{1,1},{2,3}}
    {{1,2},{1,3}}
   {{1},{1},{2,3}}
   {{1},{2},{1,3}}
   {{1},{1,2},{3}}
   {{1,1},{2},{3}}
  {{1},{1},{2},{3}}

Cf. A001055, A056239, A063834, A074206, A290103, A296150, A316223, A317545, A317546, A318559, A319002, A319003.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
lix[n_]:=LCM@@PrimePi/@If[n==1,{},FactorInteger[n]][[All,1]];
Table[Length[Select[Join@@Permutations/@facs[n],OrderedQ[lix/@#]&]],{n,100}]

is_weakly_increasing(v) = { for(i=2,#v,if(v[i]A290103(n) = lcm(apply(p->primepi(p),factor(n)[,1]));
A319004aux(n, facs) = if(1==n, is_weakly_increasing(apply(f -> A290103(f),Vec(facs))), my(s=0, newfacs); fordiv(n, d, if((d>1), newfacs = List(facs); listput(newfacs,d); s += A319004aux(n/d, newfacs))); (s));
A319004(n) = if((1==n)||isprime(n),1,A319004aux(n, List([]))); \\ Antti Karttunen, Sep 23 2018

A001055(n) <= a(n) <= A074206(n). - Antti Karttunen, Sep 23 2018

More terms from Antti Karttunen, Sep 23 2018

A319118 Number of multimin tree-factorizations of n.

Original entry on oeis.org

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, 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, 568, 2, 8, 1, 8, 2, 8, 1, 268, 1, 2, 6, 8, 2, 8, 1, 244, 24

Offset: 1

Gus Wiseman, Sep 10 2018

nonn

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 a(12) = 8 multimin tree-factorizations:
  12,
  (2*6), (4*3), (6*2), (2*2*3),
  (2*(2*3)), ((2*2)*3), ((2*3)*2).
Or as series-reduced plane trees of multisets:
  112,
  (1,12), (11,2), (12,1), (1,1,2),
  (1,(1,2)), ((1,1),2), ((1,2),1).

Cf. A001055, A005804, A020639, A118376, A196545, A255397, A281113, A281118, A281119, A295279, A317545, A317546, A318577, 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[Length[mmftrees[n]],{n,100}]

a(prime^n) = A118376(n).

a(product of n distinct primes) = A005804(n).

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

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 10 2018

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

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A319004 Number of ordered factorizations of n where the sequence of LCMs of the prime indices (A290103) of each factor is weakly increasing.

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A319118 Number of multimin tree-factorizations of n.

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

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

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