cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A318577 Number of complete multimin tree-factorizations of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 11, 1, 3, 1, 4, 1, 1, 1, 19, 1, 1, 3, 4, 1, 4, 1, 45, 1, 1, 1, 17, 1, 1, 1, 19, 1, 4, 1, 4, 4, 1, 1, 96, 1, 3, 1, 4, 1, 11, 1, 19, 1, 1, 1, 26, 1, 1, 4, 197, 1, 4, 1, 4, 1, 4, 1, 104, 1, 1, 3, 4, 1, 4, 1, 96, 11, 1, 1, 26, 1, 1, 1, 19, 1, 19, 1, 4, 1, 1, 1, 501, 1, 3, 4, 17
Offset: 1

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Author

Gus Wiseman, Sep 11 2018

Keywords

Comments

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.

Examples

			The a(12) = 4 trees are (2*2*3), (2*(2*3)), ((2*3)*2), ((2*2)*3).

Crossrefs

Cf. A000311, A001003, A001055, A020639, A255397, A281113, A281118, A281119, A295281, A317545, A319118, A319121.

Programs

  • Mathematica
    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[Select[mmftrees[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,100}]

Formula

a(prime^n) = A001003(n - 1).
a(product of n distinct primes) = A000311(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

Views

Author

Gus Wiseman, Sep 10 2018

Keywords

Comments

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).

Examples

			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).

Crossrefs

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

Programs

  • Mathematica
    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}]

Extensions

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

Views

Author

Gus Wiseman, Sep 11 2018

Keywords

Comments

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).

Examples

			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).

Crossrefs

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

Programs

  • Mathematica
    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}]

Extensions

a(11)-a(12) from Robert Price, Sep 14 2018
Showing 1-3 of 3 results.

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