A322453 -id:A322453 - cp's OEIS frontend

A336424 Number of factorizations of n where each factor belongs to A130091 (numbers with distinct prime multiplicities).

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 5, 2, 1, 3, 3, 1, 1, 1, 7, 1, 1, 1, 6, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 9, 2, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 4, 1, 1, 3, 11, 1, 1, 1, 3, 1, 1, 1, 11, 1, 1, 3, 3, 1, 1, 1, 9, 5, 1, 1, 4, 1, 1

Offset: 1

Gus Wiseman, Aug 03 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			The a(n) factorizations for n = 2, 4, 8, 60, 16, 36, 32, 48:
  2  4    8      5*12     16       4*9      32         48
     2*2  2*4    3*20     4*4      3*12     4*8        4*12
          2*2*2  3*4*5    2*8      3*3*4    2*16       3*16
                 2*2*3*5  2*2*4    2*18     2*4*4      3*4*4
                          2*2*2*2  2*2*9    2*2*8      2*24
                                   2*2*3*3  2*2*2*4    2*3*8
                                            2*2*2*2*2  2*2*12
                                                       2*2*3*4
                                                       2*2*2*2*3

A327523 is the case when n is restricted to belong to A130091 also.
A001055 counts factorizations.
A007425 counts divisors of divisors.
A045778 counts strict factorizations.
A074206 counts ordered factorizations.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts nonempty chains of divisors.
A281116 counts factorizations with no common divisor.
A302696 lists numbers whose prime indices are pairwise coprime.
A305149 counts stable factorizations.
A320439 counts factorizations using A289509.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336500 counts divisors of n in A130091 with quotient also in A130091.
A336568 = not a product of two numbers with distinct prime multiplicities.
A336569 counts maximal chains of elements of A130091.
A337256 counts chains of divisors.
Cf. A071625, A080688, A098859, A118914, A124010, A167865, A294068, A303707, A305150, A322453, A336420, A336570, A336571.

facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Table[Length[facsusing[Select[Range[2,n],UnsameQ@@Last/@FactorInteger[#]&],n]],{n,100}]

A327012 Number of factorizations of n into factors > 1 whose dual is a (strict) antichain.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1

Offset: 1

Gus Wiseman, Aug 13 2019

nonn

Differs from A322453 at 36, 72, 100, ...
The dual of a multiset system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The dual of a factorization is the dual of the multiset partition obtained by replacing each factor with its multiset of prime indices.
An antichain is a set of multisets, none of which is a submultiset of any other.

			The a(72) = 12 factorizations:
  (8*9)
  (3*24)
  (4*18)
  (2*4*9)
  (3*3*8)
  (3*4*6)
  (2*2*18)
  (2*3*12)
  (2*2*2*9)
  (2*2*3*6)
  (2*3*3*4)
  (2*2*2*3*3)

Set-systems whose dual is a (strict) antichain are A326965.
The version where the dual is a weak antichain is A326975.
Partitions whose dual is a (strict) antichain are A326977.
Cf. A001055, A316978, A326961, A326974, A326976, A326979.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[facs[n],UnsameQ@@dual[primeMS/@#]&&stableQ[dual[primeMS/@#],submultQ]&]],{n,100}]

cp's OEIS Frontend

A336424 Number of factorizations of n where each factor belongs to A130091 (numbers with distinct prime multiplicities).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica