A317791 Number of non-isomorphic multiset partitions of the multiset of prime indices of n (row n of A112798).

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 3, 1, 7, 2, 2, 2, 7, 1, 2, 2, 7, 1, 3, 1, 4, 4, 2, 1, 12, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 9, 1, 2, 4, 11, 2, 3, 1, 4, 2, 3, 1, 16, 1, 2, 4, 4, 2, 3, 1, 12, 5, 2, 1, 9, 2, 2, 2

Offset: 1

Author

Gus Wiseman, Aug 07 2018

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n.

a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Dec 03 2018

Are any terms of the complement known? In particular, does this sequence contain 6? - Gus Wiseman, Oct 21 2022

Examples

			Non-isomorphic representatives of the a(42) = 3 multiset partitions are {{1,2,4}}, {{1},{2,4}}, {{1},{2},{4}}.
Non-isomorphic representatives of the a(60) = 9 multiset partitions:
  {1123},
  {1}{123}, {2}{113}, {11}{23}, {12}{13},
  {1}{1}{23}, {1}{2}{13}, {2}{3}{11},
  {1}{1}{2}{3}.
Missing from this list are {3}{112} and {1}{3}{12}, which are isomorphic to {2}{113} and {1}{2}{13} respectively.
For n = 180 = 2^2 * 3^2 * 5, there are A001055(180) = 26 different factorizations to one or more factors larger than 1. Of these 18 are such that by swapping 2 and 3 in each factor of that factorization the result is another, different factorization of 180, while the other 8 cases are such that 2 <-> 3 swap doesn't change the factorization. Thus a(180) = 18/2 + 8 = 17. - _Antti Karttunen_, Dec 03 2018
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Gus Wiseman)
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A001055, A007716, A045778, A056239, A112798, A281116, A317533, A317757, A318285.

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]]]];
sysnorm[{}] := {};sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[sysnorm/@mps[primeMS[n]]]],{n,100}]

Formula

For all n, a(n) <= A001055(n). - Antti Karttunen, Dec 01 2018

If n is squarefree with k prime factors, or if n = p^k for p prime, we have a(n) = A000041(k).

a(n) = A318285(A181819(n)). - Andrew Howroyd, Jan 17 2023

Extensions

Terms corrected by Gus Wiseman, Dec 04 2018