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 1-4 of 4 results.

A294788 Number of twice-factorizations of type (Q,P,Q) and product n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 5, 1, 3, 3, 3, 1, 5, 1, 5, 3, 3, 1, 12, 1, 3, 3, 5, 1, 12, 1, 5, 3, 3, 3, 13, 1, 3, 3, 12, 1, 12, 1, 5, 5, 3, 1, 19, 1, 5, 3, 5, 1, 12, 3, 12, 3, 3, 1, 26, 1, 3, 5, 11, 3, 12, 1, 5, 3, 12, 1, 26, 1, 3, 5, 5, 3, 12, 1, 19, 3, 3
Offset: 1

Views

Author

Gus Wiseman, Nov 08 2017

Keywords

Comments

a(n) is the number of ways to choose a product-preserving permutation of a set partition of a factorization of n into distinct factors greater than one.

Examples

			The a(36) = 13 twice-factorizations are: (2)*(3)*(6), (2)*(3*6), (6)*(2*3), (2*3)*(6), (2*6)*(3), (2*3*6), (2)*(18), (2*18), (3)*(12), (3*12), (4)*(9), (4*9), (36).

Crossrefs

Cf. A001055, A063834, A279790, A281113, A294786, A294787.

Programs

  • Mathematica
    nn=100;
sfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sfs[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,_}];
Table[Total[Sum[Times@@Factorial/@Length/@Split[Sort[Times@@@f]],{f,sps[Sort[#]]}]&/@sfs[n]],{n,nn}]

A294786 Number of ways to choose a set partition of a factorization of n into distinct factors greater than one such that different blocks have different products.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 5, 1, 3, 3, 3, 1, 5, 1, 5, 3, 3, 1, 12, 1, 3, 3, 5, 1, 12, 1, 5, 3, 3, 3, 11, 1, 3, 3, 12, 1, 12, 1, 5, 5, 3, 1, 19, 1, 5, 3, 5, 1, 12, 3, 12, 3, 3, 1, 26, 1, 3, 5, 9, 3, 12, 1, 5, 3, 12, 1, 26, 1, 3, 5, 5, 3, 12, 1, 19, 3, 3
Offset: 1

Views

Author

Gus Wiseman, Nov 08 2017

Keywords

Examples

			The a(36)=11 ways are:
(2)*(3)*(6),
(2)*(3*6), (2*6)*(3), (2)*(18), (3)*(12), (4)*(9),
(2*3*6), (2*18), (3*12), (4*9), (36).

Links

Crossrefs

Cf. A000258, A001055, A045778, A279375, A281113, A294787, A294788.

Programs

  • Mathematica
    sfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sfs[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,_}];
Table[Length[Join@@Function[fac,Select[sps[fac],UnsameQ@@Times@@@#&]]/@sfs[n]],{n,100}]

Formula

a(product of n distinct primes) = A000258(n).
a(prime^n) = A279375(n).

A320886 Number of multiset partitions of integer partitions of n where all parts have the same product.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 25, 33, 54, 73, 107, 140, 207, 264, 369, 479, 652, 828, 1112, 1400, 1848, 2326, 3009, 3762, 4856, 6020, 7648, 9478, 11942, 14705, 18427, 22576, 28083, 34350, 42429, 51714, 63680, 77289, 94618, 114648, 139773, 168799, 205144, 247128, 299310, 359958, 434443, 521255, 627812, 751665, 902862
Offset: 0

Views

Author

Gus Wiseman, Oct 23 2018

Keywords

Examples

			The a(1) = 1 through a(6) = 25 multiset partitions:
  (1)  (2)     (3)        (4)           (5)              (6)
       (11)    (12)       (13)          (14)             (15)
       (1)(1)  (111)      (22)          (23)             (24)
               (1)(11)    (112)         (113)            (33)
               (1)(1)(1)  (1111)        (122)            (114)
                          (2)(2)        (1112)           (123)
                          (1)(111)      (11111)          (222)
                          (11)(11)      (2)(12)          (1113)
                          (1)(1)(11)    (1)(1111)        (1122)
                          (1)(1)(1)(1)  (11)(111)        (3)(3)
                                        (1)(1)(111)      (11112)
                                        (1)(11)(11)      (111111)
                                        (1)(1)(1)(11)    (12)(12)
                                        (1)(1)(1)(1)(1)  (2)(112)
                                                         (2)(2)(2)
                                                         (1)(11111)
                                                         (11)(1111)
                                                         (111)(111)
                                                         (1)(1)(1111)
                                                         (1)(11)(111)
                                                         (11)(11)(11)
                                                         (1)(1)(1)(111)
                                                         (1)(1)(11)(11)
                                                         (1)(1)(1)(1)(11)
                                                         (1)(1)(1)(1)(1)(1)

Crossrefs

Cf. A001055, A001970, A045778, A050336, A279375, A294617, A294786, A294787, A294788, A320887, A320888, A320889.

Programs

  • Mathematica
    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[Length[Select[Join@@mps/@IntegerPartitions[n],SameQ@@Times@@@#&]],{n,8}]
  • PARI
    G(n)={my(M=Map()); for(k=1, n, forpart(p=k, my(t=vecprod(Vec(p)), z); mapput(M, t, if(mapisdefined(M, t, &z), z, 0) + x^k))); M}
a(n)=if(n==0, 1, vecsum(apply(p->EulerT(Vecrev(p/x, n))[n], Mat(G(n))[,2]))) \\ Andrew Howroyd, Oct 26 2018

Extensions

a(13)-a(50) from Andrew Howroyd, Oct 26 2018

A320889 Number of set partitions of strict factorizations of n into factors > 1 such that all the blocks have the same product.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 5, 1, 2, 2, 3, 1, 5, 1, 3, 2, 2, 2, 6, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 1, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 5, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 2, 2, 1, 9, 2, 2, 2
Offset: 1

Views

Author

Gus Wiseman, Oct 23 2018

Keywords

Examples

			The a(144) = 17 set partitions:
  (2*3*4*6)    (2*8*9)     (2*72)  (144)
  (2*6)*(3*4)  (3*6*8)     (3*48)
               (2*3*24)    (4*36)
               (2*4*18)    (6*24)
               (2*6*12)    (8*18)
               (3*4*12)    (9*16)
               (2*6)*(12)
               (3*4)*(12)

Crossrefs

Cf. A000110, A001055, A001970, A045778, A050336, A279375, A294786, A294787, A296132, A320886, A320887, A320888.

Programs

  • Mathematica
    strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[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,_}];
Table[Length[Join@@Table[Select[sps[fac],SameQ@@Times@@@#&],{fac,strfacs[n]}]],{n,100}]
Showing 1-4 of 4 results.

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