A330665 - OEIS

%I #4 Dec 29 2019 08:44:00

%S 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,2,1,2,1,2,1,1,1,5,1,1,1,2,1,3,1,5,1,1,

%T 1,7,1,1,1,5,1,3,1,2,2,1,1,16,1,2,1,2,1,5,1,5,1,1,1,11,1,1,2,16,1,3,1,

%U 2,1,3,1,27,1,1,2,2,1,3,1,16,2,1,1,11,1

%N Number of balanced reduced multisystems of maximal depth whose atoms are the prime indices of n.

%C First differs from A317145 at a(32) = 5, A317145(32) = 4.

%C A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

%C 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.

%C Also series/singleton-reduced factorizations of n with Omega(n) levels of parentheses. See A001055, A050336, A050338, A050340, etc.

%F a(2^n) = A000111(n - 1).

%F a(product of n distinct primes) = A006472(n).

%e The a(n) multisystems for n = 2, 6, 12, 24, 48:

%e   {1}  {1,2}  {{1},{1,2}}  {{{1}},{{1},{1,2}}}  {{{{1}}},{{{1}},{{1},{1,2}}}}

%e               {{2},{1,1}}  {{{1,1}},{{1},{2}}}  {{{{1}}},{{{1,1}},{{1},{2}}}}

%e                            {{{1}},{{2},{1,1}}}  {{{{1},{1}}},{{{1}},{{1,2}}}}

%e                            {{{1,2}},{{1},{1}}}  {{{{1},{1,1}}},{{{1}},{{2}}}}

%e                            {{{2}},{{1},{1,1}}}  {{{{1,1}}},{{{1}},{{1},{2}}}}

%e                                                 {{{{1}}},{{{1}},{{2},{1,1}}}}

%e                                                 {{{{1}}},{{{1,2}},{{1},{1}}}}

%e                                                 {{{{1},{1}}},{{{2}},{{1,1}}}}

%e                                                 {{{{1},{1,2}}},{{{1}},{{1}}}}

%e                                                 {{{{1,1}}},{{{2}},{{1},{1}}}}

%e                                                 {{{{1}}},{{{2}},{{1},{1,1}}}}

%e                                                 {{{{1},{2}}},{{{1}},{{1,1}}}}

%e                                                 {{{{1,2}}},{{{1}},{{1},{1}}}}

%e                                                 {{{{2}}},{{{1}},{{1},{1,1}}}}

%e                                                 {{{{2}}},{{{1,1}},{{1},{1}}}}

%e                                                 {{{{2},{1,1}}},{{{1}},{{1}}}}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

%t totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1<Length[#]<Length[m]&]}],m];

%t Table[Length[Select[totm[primeMS[n]],Length[#]<=1||Depth[#]==PrimeOmega[n]&]],{n,100}]

%Y The last nonzero term in row n of A330667 is a(n).

%Y The chain version is A317145.

%Y The non-maximal version is A318812.

%Y Unlabeled versions are A330664 and A330663.

%Y Other labeled versions are A330675 (strongly normal) and A330676 (normal).

%Y Cf. A001055, A005121, A005804, A050336, A213427, A292505, A317144, A318849, A320160, A330474, A330475, A330679.

%K nonn

%O 1,12

%A _Gus Wiseman_, Dec 27 2019