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Numbers whose number of prime factors with multiplicity (A001222) is the number of distinct prime factors (A001221) in the product of the prime indices (A003963).
13

%I #26 Dec 18 2018 17:07:29

%S 1,3,5,7,11,15,17,19,23,26,31,33,35,39,41,51,53,55,58,59,65,67,69,74,

%T 77,83,85,86,87,91,93,94,95,97,103,109,111,119,122,123,127,129,131,

%U 142,146,155,157,158,161,165,169,177,178,179,183,185,187,191,201,202

%N Numbers whose number of prime factors with multiplicity (A001222) is the number of distinct prime factors (A001221) in the product of the prime indices (A003963).

%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 square multiset multisystems, meaning the number of edges is equal to the number of distinct vertices.

%e The sequence of multiset multisystems whose MM-numbers belong to the sequence begins:

%e 1: {}

%e 3: {{1}}

%e 5: {{2}}

%e 7: {{1,1}}

%e 11: {{3}}

%e 15: {{1},{2}}

%e 17: {{4}}

%e 19: {{1,1,1}}

%e 23: {{2,2}}

%e 26: {{},{1,2}}

%e 31: {{5}}

%e 33: {{1},{3}}

%e 35: {{2},{1,1}}

%e 39: {{1},{1,2}}

%e 41: {{6}}

%e 51: {{1},{4}}

%e 53: {{1,1,1,1}}

%e 55: {{2},{3}}

%e 58: {{},{1,3}}

%e 59: {{7}}

%e 65: {{2},{1,2}}

%e 67: {{8}}

%e 69: {{1},{2,2}}

%e 74: {{},{1,1,2}}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],PrimeOmega[#]==PrimeNu[Times@@primeMS[#]]&]

%Y Cf. A003963, A057151, A064573, A120732, A319616, A319877, A320325, A322527, A322530.

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 17 2018