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%I #5 Jan 08 2024 14:30:42
%S 1,3,5,9,11,13,15,17,25,27,29,31,33,39,41,43,45,47,51,55,59,65,67,73,
%T 75,79,81,83,85,87,93,99,101,109,113,117,121,123,125,127,129,135,137,
%U 139,141,143,145,149,153,155,157,163,165,167,169,177,179,181,187
%N Products of odd primes of squarefree index. MM-numbers of set multipartitions.
%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 of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
%C A set multipartition is a finite multiset of finite nonempty sets.
%e The terms together with the corresponding set multipartitions begin:
%e 1: {}
%e 3: {{1}}
%e 5: {{2}}
%e 9: {{1},{1}}
%e 11: {{3}}
%e 13: {{1,2}}
%e 15: {{1},{2}}
%e 17: {{4}}
%e 25: {{2},{2}}
%e 27: {{1},{1},{1}}
%e 29: {{1,3}}
%e 31: {{5}}
%e 33: {{1},{3}}
%e 39: {{1},{1,2}}
%e 41: {{6}}
%e 43: {{1,4}}
%e 45: {{1},{1},{2}}
%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[100],OddQ[#]&&And@@SquareFreeQ/@prix[#]&]
%Y Odd case of A302478.
%Y Products of odd terms of A302491.
%Y A049311 counts non-isomorphic set multipartitions, strict A283877.
%Y A050320 counts set multipartitions of prime indices.
%Y A056239 adds up prime indices, row sums of A112798.
%Y A089259 counts set multipartitions of integer partitions.
%Y A116540 counts set multipartitions covering an initial interval by weight.
%Y A368533 lists numbers with squarefree binary indices.
%Y Cf. A000040, A000720, A001222, A005117, A006450, A076610, A270995, A296119, A302242, A302590, A339113.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jan 08 2024
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