A356931 Number of multiset partitions of the prime indices of n into multisets of odd numbers. Number of factorizations of n into members of A066208.

1, 1, 0, 2, 1, 0, 0, 3, 0, 2, 1, 0, 0, 0, 0, 5, 1, 0, 0, 4, 0, 2, 1, 0, 2, 0, 0, 0, 0, 0, 1, 7, 0, 2, 0, 0, 0, 0, 0, 7, 1, 0, 0, 4, 0, 2, 1, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 2, 0, 11, 0, 0, 1, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 12, 0, 2, 1, 0, 2, 0

Offset

1,4

Comments

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..86.

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

Formula

a(n) = 0 if n is in A324929, otherwise a(n) = A001055(n).

Example

The a(440) = 21 multiset partitions of {1,1,1,3,5}:
  {1}{1}{1}{3}{5}  {1}{1}{1}{35}  {1}{1}{135}  {1}{1135}  {11135}
                   {1}{1}{13}{5}  {1}{11}{35}  {11}{135}
                   {1}{11}{3}{5}  {11}{13}{5}  {111}{35}
                   {1}{1}{3}{15}  {1}{13}{15}  {113}{15}
                                  {11}{3}{15}  {13}{115}
                                  {1}{3}{115}  {3}{1115}
                                  {1}{5}{113}  {5}{1113}
                                  {3}{111}{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]]}]];
Table[Length[Select[facs[n], And@@(OddQ[Times@@primeMS[#]]&/@#)&]], {n, 100}]

Crossrefs

Positions of 0's are A324929, complement A066208.

A000688 counts factorizations into prime powers.

A001055 counts factorizations.

A001221 counts prime divisors, sum A001414.

A001222 counts prime factors with multiplicity.

A056239 adds up prime indices, row sums of A112798.

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

Other types: A000009, A000012, A000041, A089259, A356930.

Other conditions: A050320, A050330, A356936, A322585, A356233, A356945.

Cf. A003963, A073491, A107742, A287170, A325534, A325535, A340852, A356234.

Sequence in context: A364022 A363900 A209777 * A356864 A145677 A128229

Adjacent sequences: A356928 A356929 A356930 * A356932 A356933 A356934

Keyword

nonn

Author

Gus Wiseman, Sep 08 2022

Status

approved