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%I #4 Jun 21 2019 22:46:08
%S 49,91,98,133,147,169,182,196,203,245,247,259,266,273,294,299,301,338,
%T 343,361,364,371,377,392,399,406,427,441,455,481,490,494,497,507,518,
%U 529,532,539,546,551,553,559,588,598,602,609,623,637,665,667,676,686,689
%N MM-numbers of weakly nesting multiset partitions.
%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. The multiset multisystem with MM-number n is obtained by taking the multiset of prime indices of each prime index of n.
%C A multiset partition is weakly nesting if it has two blocks of the form {...x,y...}, {...z,t...} where x <= z and t <= y or z <= x and y <= t.
%e The sequence of terms together with their multiset multisystems begins:
%e 49: {{1,1},{1,1}}
%e 91: {{1,1},{1,2}}
%e 98: {{},{1,1},{1,1}}
%e 133: {{1,1},{1,1,1}}
%e 147: {{1},{1,1},{1,1}}
%e 169: {{1,2},{1,2}}
%e 182: {{},{1,1},{1,2}}
%e 196: {{},{},{1,1},{1,1}}
%e 203: {{1,1},{1,3}}
%e 245: {{2},{1,1},{1,1}}
%e 247: {{1,2},{1,1,1}}
%e 259: {{1,1},{1,1,2}}
%e 266: {{},{1,1},{1,1,1}}
%e 273: {{1},{1,1},{1,2}}
%e 294: {{},{1},{1,1},{1,1}}
%e 299: {{1,2},{2,2}}
%e 301: {{1,1},{1,4}}
%e 338: {{},{1,2},{1,2}}
%e 343: {{1,1},{1,1},{1,1}}
%e 361: {{1,1,1},{1,1,1}}
%t wknXQ[stn_]:=MatchQ[stn,{___,{___,x_,y_,___},___,{___,z_,t_,___},___}/;(x<=z&&y>=t)||(x>=z&&y<=t)]
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[1000],wknXQ[primeMS/@primeMS[#]]&]
%Y MM-numbers of crossing multiset partitions are A324170.
%Y MM-numbers of nesting multiset partitions are A324256.
%Y MM-numbers of capturing multiset partitions are A326255.
%Y Nesting set partitions are A016098.
%Y Cf. A001055, A034827, A056239, A112798, A122880, A302242, A324171.
%Y Cf. A326211, A326243, A326258, A326260.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jun 21 2019
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