A356936 Number of multiset partitions of the multiset of prime indices of n into intervals. Number of factorizations of n into members of A073485.

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset

1,6

Comments

An interval is a set of positive integers with all differences of adjacent elements equal to 1.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Links

Table of n, a(n) for n=1..87.

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Example

The a(n) multiset partitions for n = 6, 30, 36, 90, 180:
  {12}    {123}      {12}{12}      {12}{23}      {12}{123}
  {1}{2}  {1}{23}    {1}{2}{12}    {2}{123}      {1}{12}{23}
          {3}{12}    {1}{1}{2}{2}  {1}{2}{23}    {1}{2}{123}
          {1}{2}{3}                {2}{3}{12}    {3}{12}{12}
                                   {1}{2}{2}{3}  {1}{1}{2}{23}
                                                 {1}{2}{3}{12}
                                                 {1}{1}{2}{2}{3}
The a(n) factorizations for n = 6, 30, 36, 90, 180:
  (6)    (30)     (6*6)      (3*30)     (6*30)
  (2*3)  (5*6)    (2*3*6)    (6*15)     (5*6*6)
         (2*15)   (2*2*3*3)  (3*5*6)    (2*3*30)
         (2*3*5)             (2*3*15)   (2*6*15)
                             (2*3*3*5)  (2*3*5*6)
                                        (2*2*3*15)
                                        (2*2*3*3*5)

Mathematica

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]]}]];
chQ[y_]:=Or[Length[y]<=1, Union[Differences[y]]=={1}];
Table[Length[Select[facs[n], And@@chQ/@primeMS/@#&]], {n, 100}]

Crossrefs

A000688 counts factorizations into prime powers.

A001055 counts factorizations.

A001221 counts prime divisors, sum A001414.

A001222 counts prime factors with multiplicity.

A356069 counts gapless divisors, initial A356224 (complement A356225).

A056239 adds up prime indices, row sums of A112798.

Intervals are counted by A000012, A001227, ranked by A073485.

Other types: A107742, A356233, A356937, A356938, A356939.

Other conditions: A050320, A050330, A322585, A356931, A356945.

Cf. A003963, A073491, A287170, A328195, A356234.

Sequence in context: A280801 A085006 A300820 * A257990 A369257 A257743

Adjacent sequences: A356933 A356934 A356935 * A356937 A356938 A356939

Keyword

nonn

Author

Gus Wiseman, Sep 08 2022

Status

approved