cp's OEIS Frontend

A330665 Number of balanced reduced multisystems of maximal depth whose atoms are the prime indices of n.

%I A330665 #4 Dec 29 2019 08:44:00
%S A330665 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 A330665 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 A330665 2,1,3,1,27,1,1,2,2,1,3,1,16,2,1,1,11,1
%N A330665 Number of balanced reduced multisystems of maximal depth whose atoms are the prime indices of n.
%C A330665 First differs from A317145 at a(32) = 5, A317145(32) = 4.
%C A330665 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 A330665 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 A330665 Also series/singleton-reduced factorizations of n with Omega(n) levels of parentheses. See A001055, A050336, A050338, A050340, etc.
%F A330665 a(2^n) = A000111(n - 1).
%F A330665 a(product of n distinct primes) = A006472(n).
%e A330665 The a(n) multisystems for n = 2, 6, 12, 24, 48:
%e A330665   {1}  {1,2}  {{1},{1,2}}  {{{1}},{{1},{1,2}}}  {{{{1}}},{{{1}},{{1},{1,2}}}}
%e A330665               {{2},{1,1}}  {{{1,1}},{{1},{2}}}  {{{{1}}},{{{1,1}},{{1},{2}}}}
%e A330665                            {{{1}},{{2},{1,1}}}  {{{{1},{1}}},{{{1}},{{1,2}}}}
%e A330665                            {{{1,2}},{{1},{1}}}  {{{{1},{1,1}}},{{{1}},{{2}}}}
%e A330665                            {{{2}},{{1},{1,1}}}  {{{{1,1}}},{{{1}},{{1},{2}}}}
%e A330665                                                 {{{{1}}},{{{1}},{{2},{1,1}}}}
%e A330665                                                 {{{{1}}},{{{1,2}},{{1},{1}}}}
%e A330665                                                 {{{{1},{1}}},{{{2}},{{1,1}}}}
%e A330665                                                 {{{{1},{1,2}}},{{{1}},{{1}}}}
%e A330665                                                 {{{{1,1}}},{{{2}},{{1},{1}}}}
%e A330665                                                 {{{{1}}},{{{2}},{{1},{1,1}}}}
%e A330665                                                 {{{{1},{2}}},{{{1}},{{1,1}}}}
%e A330665                                                 {{{{1,2}}},{{{1}},{{1},{1}}}}
%e A330665                                                 {{{{2}}},{{{1}},{{1},{1,1}}}}
%e A330665                                                 {{{{2}}},{{{1,1}},{{1},{1}}}}
%e A330665                                                 {{{{2},{1,1}}},{{{1}},{{1}}}}
%t A330665 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A330665 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A330665 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A330665 totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1<Length[#]<Length[m]&]}],m];
%t A330665 Table[Length[Select[totm[primeMS[n]],Length[#]<=1||Depth[#]==PrimeOmega[n]&]],{n,100}]
%Y A330665 The last nonzero term in row n of A330667 is a(n).
%Y A330665 The chain version is A317145.
%Y A330665 The non-maximal version is A318812.
%Y A330665 Unlabeled versions are A330664 and A330663.
%Y A330665 Other labeled versions are A330675 (strongly normal) and A330676 (normal).
%Y A330665 Cf. A001055, A005121, A005804, A050336, A213427, A292505, A317144, A318849, A320160, A330474, A330475, A330679.
%K A330665 nonn
%O A330665 1,12
%A A330665 _Gus Wiseman_, Dec 27 2019