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%I #5 Jan 05 2020 08:11:03
%S 1,1,1,1,2,1,3,2,1,3,1,7,7,1,5,5,1,5,9,5,1,9,11,1,9,28,36,16,1,10,24,
%T 16,1,14,38,27,1,13,18,1,13,69,160,164,61,1,24,79,62,1,20,160,580,
%U 1022,855,272,1,19,59,45,1,27,138,232,123,1,17,77,121,61
%N Irregular triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k whose degrees (atom multiplicities) are the prime indices of n.
%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. A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
%F T(2^n,k) = A008826(n,k).
%e Triangle begins:
%e {}
%e 1
%e 1
%e 1 1
%e 1 2
%e 1 3 2
%e 1 3
%e 1 7 7
%e 1 5 5
%e 1 5 9 5
%e 1 9 11
%e 1 9 28 36 16
%e 1 10 24 16
%e 1 14 38 27
%e 1 13 18
%e 1 13 69 160 164 61
%e 1 24 79 62
%e For example, row n = 12 counts the following multisystems:
%e {1,1,2,3} {{1},{1,2,3}} {{{1}},{{1},{2,3}}}
%e {{1,1},{2,3}} {{{1,1}},{{2},{3}}}
%e {{1,2},{1,3}} {{{1}},{{2},{1,3}}}
%e {{2},{1,1,3}} {{{1,2}},{{1},{3}}}
%e {{3},{1,1,2}} {{{1}},{{3},{1,2}}}
%e {{1},{1},{2,3}} {{{1,3}},{{1},{2}}}
%e {{1},{2},{1,3}} {{{2}},{{1},{1,3}}}
%e {{1},{3},{1,2}} {{{2}},{{3},{1,1}}}
%e {{2},{3},{1,1}} {{{2,3}},{{1},{1}}}
%e {{{3}},{{1},{1,2}}}
%e {{{3}},{{2},{1,1}}}
%t nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[Reverse[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[nrmptn[n]],Depth[#]==k&]],{n,2,10},{k,2,Length[nrmptn[n]]}]
%Y Row sums are A318846.
%Y Final terms in each row are A330728.
%Y Row prime(n) is row n of A330784.
%Y Row 2^n is row n of A008826.
%Y Row n is row A181821(n) of A330667.
%Y Column k = 3 is A318284(n) - 2 for n > 2.
%Y Cf. A000111, A002846, A005121, A292504, A318812, A318813, A318847, A318848, A318849, A330475, A330666, A330935.
%K nonn,tabf
%O 2,5
%A _Gus Wiseman_, Jan 04 2020
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