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A356931

Number of multiset partitions of the prime indices of n into multisets of odd numbers. Number of factorizations of n into members ofA066208.

3

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

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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 ofA112798.

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 inA324929, 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 areA324929, complementA066208.

A000688 counts factorizations into prime powers.

A001055 counts factorizations.

A001221 counts prime divisors, sumA001414.

A001222 counts prime factors with multiplicity.

A056239 adds up prime indices, row sums ofA112798.

A356069 counts gapless divisors, initialA356224 (complementA356225).

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 *A377129 A356864 A145677

Adjacent sequences:A356928 A356929 A356930 *A356932 A356933 A356934

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 08 2022

STATUS

approved

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