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%I #43 Dec 18 2018 17:07:23
%S 1,7,9,23,25,97,121,151,161,169,175,183,185,195,207,225,227,289,541,
%T 661,679,687,781,841,847,873,957,961,1009,1089,1193,1427,1563,1589,
%U 1681,1819,1849,1879,1895,2023,2043,2167,2193,2209,2231,2425,2437,2585,2601
%N Odd numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).
%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 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 multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of 2-regular (all vertex-degrees are 2) multiset partitions (no empty parts).
%e The sequence of multiset partitions whose MM-numbers belong to the sequence begins:
%e 1: {}
%e 7: {{1,1}}
%e 9: {{1},{1}}
%e 23: {{2,2}}
%e 25: {{2},{2}}
%e 97: {{3,3}}
%e 121: {{3},{3}}
%e 151: {{1,1,2,2}}
%e 161: {{1,1},{2,2}}
%e 169: {{1,2},{1,2}}
%e 175: {{2},{2},{1,1}}
%e 183: {{1},{1,2,2}}
%e 185: {{2},{1,1,2}}
%e 195: {{1},{2},{1,2}}
%e 207: {{1},{1},{2,2}}
%e 225: {{1},{1},{2},{2}}
%e 227: {{4,4}}
%e 289: {{4},{4}}
%e 541: {{1,1,3,3}}
%e 661: {{5,5}}
%e 679: {{1,1},{3,3}}
%e 687: {{1},{1,3,3}}
%e 781: {{3},{1,1,3}}
%e 841: {{1,3},{1,3}}
%e 847: {{1,1},{3},{3}}
%e 873: {{1},{1},{3,3}}
%e 957: {{1},{3},{1,3}}
%e 961: {{5},{5}}
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[1,100,2],Or[#==1,SameQ[##,2]&@@Last/@FactorInteger[Times@@primeMS[#]]]&]
%Y Cf. A003963, A005117, A005176, A062503, A064573, A072774, A295193, A302505, A319877, A319899, A320325, A322526, A322527, A322530.
%K nonn
%O 1,2
%A _Gus Wiseman_, Dec 17 2018
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