cp's OEIS Frontend

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

%I A317791 #28 Jan 17 2023 21:24:42
%S A317791 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,
%T A317791 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,
%U A317791 4,2,3,1,16,1,2,4,4,2,3,1,12,5,2,1,9,2,2,2
%N A317791 Number of non-isomorphic multiset partitions of the multiset of prime indices of n (row n of A112798).
%C A317791 A prime index of n is a number m such that prime(m) divides n.
%C A317791 a(n) depends only on prime signature of n (cf. A025487).  - _Antti Karttunen_, Dec 03 2018
%C A317791 Are any terms of the complement known? In particular, does this sequence contain 6? - _Gus Wiseman_, Oct 21 2022
%H A317791 Andrew Howroyd, <a href="/A317791/b317791.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Gus Wiseman)
%H A317791 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A317791 For all n, a(n) <= A001055(n). - _Antti Karttunen_, Dec 01 2018
%F A317791 If n is squarefree with k prime factors, or if n = p^k for p prime, we have a(n) = A000041(k).
%F A317791 a(n) = A318285(A181819(n)). - _Andrew Howroyd_, Jan 17 2023
%e A317791 Non-isomorphic representatives of the a(42) = 3 multiset partitions are {{1,2,4}}, {{1},{2,4}}, {{1},{2},{4}}.
%e A317791 Non-isomorphic representatives of the a(60) = 9 multiset partitions:
%e A317791   {1123},
%e A317791   {1}{123}, {2}{113}, {11}{23}, {12}{13},
%e A317791   {1}{1}{23}, {1}{2}{13}, {2}{3}{11},
%e A317791   {1}{1}{2}{3}.
%e A317791 Missing from this list are {3}{112} and {1}{3}{12}, which are isomorphic to {2}{113} and {1}{2}{13} respectively.
%e A317791 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
%t A317791 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A317791 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A317791 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]]}]])]];
%t A317791 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A317791 Table[Length[Union[sysnorm/@mps[primeMS[n]]]],{n,100}]
%Y A317791 Cf. A001055, A007716, A045778, A056239, A112798, A281116, A317533, A317757, A318285.
%K A317791 nonn
%O A317791 1,4
%A A317791 _Gus Wiseman_, Aug 07 2018
%E A317791 Terms corrected by _Gus Wiseman_, Dec 04 2018