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 1-6 of 6 results.

A296119 Number of ways to choose a strict factorization of each factor in a factorization of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 7, 1, 3, 3, 7, 1, 7, 1, 7, 3, 3, 1, 16, 2, 3, 4, 7, 1, 12, 1, 12, 3, 3, 3, 21, 1, 3, 3, 16, 1, 12, 1, 7, 7, 3, 1, 33, 2, 7, 3, 7, 1, 16, 3, 16, 3, 3, 1, 34, 1, 3, 7, 23, 3, 12, 1, 7, 3, 12, 1, 50, 1, 3, 7, 7, 3, 12, 1, 33, 7, 3
Offset: 1

Views

Author

Gus Wiseman, Dec 05 2017

Keywords

Examples

			The a(24) = 16 twice-factorizations:
(2)*(2)*(2)*(3),
(2)*(2)*(2*3), (2)*(2)*(6), (2)*(3)*(4),
(2)*(2*6), (2)*(3*4), (2)*(12), (3)*(2*4), (3)*(8), (4)*(2*3), (4)*(6),
(2*3*4), (2*12), (3*8), (4*6), (24).

Links

Crossrefs

Cf. A000009, A005117, A045778, A270995, A281113, A294788, A296118, A296120, A296121.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Times@@(Length[Select[facs[#],UnsameQ@@#&]]&/@fac),{fac,facs[n]}],{n,100}]
  • PARI
    A045778(n, m=n) = ((n<=m) + sumdiv(n, d, if((d>1)&&(d<=m)&&(dA045778(n/d, d-1))));
A296119(n, m=n) = if(1==n, 1, sumdiv(n, d, if((d>1)&&(d<=m), A045778(d)*A296119(n/d, d)))); \\ Antti Karttunen, Oct 08 2018

Formula

Dirichlet g.f.: 1/Product_{n > 1}(1 - A045778(n)/n^s).

A296122 Number of twice-partitions of n with no repeated partitions.

Original entry on oeis.org

1, 1, 2, 5, 10, 20, 40, 77, 157, 285, 552, 1018, 1921, 3484, 6436, 11622, 21082, 37550, 67681, 119318, 211792, 372003, 653496, 1137185, 1986234, 3429650, 5935970, 10205907, 17537684, 29958671, 51189932, 86967755, 147759421, 249850696, 422123392, 710495901
Offset: 0

Views

Author

Gus Wiseman, Dec 05 2017

Keywords

Comments

a(n) is the number of sequences of distinct integer partitions whose sums are weakly decreasing and add up to n.

Examples

			The a(4) = 10 twice-partitions: (4), (31), (22), (211), (1111), (3)(1), (21)(1), (111)(1), (2)(11), (11)(2).

Links

Crossrefs

Cf. A000009, A000041, A047968, A063834, A261049, A273873, A279375, A295279, A296121.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(j!*
      binomial(combinat[numbpart](i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 06 2017
  • Mathematica
    Table[Length[Join@@Table[Select[Tuples[IntegerPartitions/@p],UnsameQ@@#&],{p,IntegerPartitions[n]}]],{n,15}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[j!*
     Binomial[PartitionsP[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := b[n, n];
a /@ Range[0, 40] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

Extensions

a(15)-a(34) from Robert G. Wilson v, Dec 06 2017

A296120 Number of ways to choose a strict factorization of each factor in a strict factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 13, 1, 3, 3, 6, 1, 12, 1, 7, 3, 3, 3, 14, 1, 3, 3, 13, 1, 12, 1, 6, 6, 3, 1, 25, 1, 6, 3, 6, 1, 13, 3, 13, 3, 3, 1, 31, 1, 3, 6, 11, 3, 12, 1, 6, 3, 12, 1, 36, 1, 3, 6, 6, 3, 12, 1, 25, 4, 3
Offset: 1

Views

Author

Gus Wiseman, Dec 05 2017

Keywords

Examples

			The a(36) = 14 twice-factorizations:
(36), (4*9), (3*12), (2*18), (2*3*6),
(4)*(9), (3)*(12), (3)*(3*4), (3)*(2*6), (2)*(18), (2)*(3*6), (2)*(2*9),
(2)*(3)*(6), (2)*(3)*(2*3).

Crossrefs

Cf. A000009, A005117, A045778, A050345, A279785, A281113, A294788, A296118, A296119, A296121.

Programs

  • Mathematica
    sfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sfs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Times@@Length/@sfs/@fac,{fac,sfs[n]}],{n,100}]

Formula

Dirichlet g.f.: Product_{n > 1}(1 + A045778(n)/n^s).

A296118 Number of ways to choose a factorization of each factor in a strict factorization of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 5, 2, 3, 1, 8, 1, 3, 3, 8, 1, 8, 1, 8, 3, 3, 1, 20, 2, 3, 5, 8, 1, 12, 1, 18, 3, 3, 3, 23, 1, 3, 3, 20, 1, 12, 1, 8, 8, 3, 1, 45, 2, 8, 3, 8, 1, 20, 3, 20, 3, 3, 1, 38, 1, 3, 8, 34, 3, 12, 1, 8, 3, 12, 1, 66, 1, 3, 8, 8, 3, 12, 1, 45, 8, 3
Offset: 1

Views

Author

Gus Wiseman, Dec 05 2017

Keywords

Examples

			The a(12) = 8 twice-factorizations are (2)*(2*3), (2)*(6), (3)*(2*2), (3)*(4), (2*2*3), (2*6), (3*4), (12).

Links

Crossrefs

Cf. A000009, A005117, A045778, A261049, A271619, A281113, A294788, A295923, A296119, A296121.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Times@@(Length[facs[#]]&/@f),{f,Select[facs[n],UnsameQ@@#&]}],{n,100}]
  • PARI
    A001055(n, m=n) = if(1==n, 1, sumdiv(n, d, if((d>1)&&(d<=m), A001055(n/d, d))));
A296118(n, m=n) = ((n<=m)*A001055(n) + sumdiv(n, d, if((d>1)&&(d<=m)&&(dA001055(d)*A296118(n/d, d-1)))); \\ Antti Karttunen, Oct 08 2018

Formula

Dirichlet g.f.: Product_{n > 1}(1 + A001055(n)/n^s).

A383310 Number of ways to choose a strict multiset partition of a factorization of n into factors > 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 5, 2, 3, 1, 8, 1, 3, 3, 9, 1, 8, 1, 8, 3, 3, 1, 20, 2, 3, 5, 8, 1, 12, 1, 19, 3, 3, 3, 24, 1, 3, 3, 20, 1, 12, 1, 8, 8, 3, 1, 46, 2, 8, 3, 8, 1, 20, 3, 20, 3, 3, 1, 38, 1, 3, 8, 37, 3, 12, 1, 8, 3, 12, 1, 67, 1, 3, 8, 8, 3, 12, 1, 46, 9, 3
Offset: 1

Views

Author

Gus Wiseman, Apr 26 2025

Keywords

Examples

			The a(36) = 24 choices:
  {{2,2,3,3}}  {{2},{2,3,3}}  {{2},{3},{2,3}}
  {{2,2,9}}    {{3},{2,2,3}}  {{2},{3},{6}}
  {{2,3,6}}    {{2,2},{3,3}}
  {{2,18}}     {{2},{2,9}}
  {{3,3,4}}    {{9},{2,2}}
  {{3,12}}     {{2},{3,6}}
  {{4,9}}      {{3},{2,6}}
  {{6,6}}      {{6},{2,3}}
  {{36}}       {{2},{18}}
               {{3},{3,4}}
               {{4},{3,3}}
               {{3},{12}}
               {{4},{9}}

Crossrefs

The case of a unique choice (positions of 1) is A008578.
This is the strict case of A050336.
For distinct strict blocks we have A050345.
For integer partitions we have A261049, strict case of A001970.
For strict blocks that are not necessarily distinct we have A296119.
Twice-partitions of this type are counted by A296122.
For normal multisets we have A317776, strict case of A255906.
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers, distinct A050326.
A281113 counts twice-factorizations, strict A296121, see A296118, A296120.
Cf. A000009, A005117, A045782, A050342, A279785, A293243, A293511, A302494, A316439, A358914, A381992, A382201.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Sum[Length[Select[mps[y],UnsameQ@@#&]],{y,facs[n]}],{n,30}]

A383311 Number of ways to choose a set multipartition (multiset of sets) of a factorization of n into factors > 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 7, 1, 3, 3, 7, 1, 7, 1, 7, 3, 3, 1, 16, 2, 3, 4, 7, 1, 12, 1, 12, 3, 3, 3, 20, 1, 3, 3, 16, 1, 12, 1, 7, 7, 3, 1, 33, 2, 7, 3, 7, 1, 16, 3, 16, 3, 3, 1, 34, 1, 3, 7, 22, 3, 12, 1, 7, 3, 12, 1, 49, 1, 3, 7, 7, 3, 12, 1, 33, 7, 3
Offset: 1

Views

Author

Gus Wiseman, Apr 28 2025

Keywords

Comments

First differs from A296119 at a(36) = 20, A296119(36) = 21.

Examples

			The a(36) = 20 choices are:
  {{2,3,6}}  {{2,3},{2,3}}  {{2},{3},{2,3}}  {{2},{2},{3},{3}}
  {{2,18}}   {{2},{2,9}}    {{2},{2},{9}}
  {{3,12}}   {{2},{3,6}}    {{2},{3},{6}}
  {{4,9}}    {{3},{2,6}}    {{3},{3},{4}}
  {{36}}     {{6},{2,3}}
             {{2},{18}}
             {{3},{3,4}}
             {{3},{12}}
             {{4},{9}}
             {{6},{6}}

Crossrefs

The case of a unique choice (positions of 1) is A008578.
For multisets of multisets we have A050336.
For sets of sets we have A050345.
For normal multisets we have A116540, strong A330783.
For integer partitions instead of factorizations we have A089259.
Twice-partitions of this type are counted by A270995.
For sets of multisets we have A383310 (distinct products A296118).
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers, distinct A050326.
A281113 counts twice-factorizations, see A294788, A296120, A296121.
A302478 gives MM-numbers of set multipartitions.
A302494 gives MM-numbers of sets of sets.
Cf. A000009, A001970, A005117, A050342, A116539, A279785, A293243, A293511, A296119, A316439, A381992, A382077.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Sum[Length[Select[mps[y], And@@UnsameQ@@@#&]], {y,facs[n]}],{n,100}]
Showing 1-6 of 6 results.

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