A319877 - OEIS

%I #40 Dec 18 2018 17:07:15

%S 1,7,9,14,18,23,25,28,36,46,50,56,72,92,97,100,112,121,144,151,161,

%T 169,175,183,184,185,194,195,200,207,224,225,227,242,288,289,302,322,

%U 338,350,366,368,370,388,390,400,414,448,450,454,484,541,576,578,604,644

%N 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 multiset multisystems (meaning all vertex-degrees are 2).

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

%e     1: {}

%e     7: {{1,1}}

%e     9: {{1},{1}}

%e    14: {{},{1,1}}

%e    18: {{},{1},{1}}

%e    23: {{2,2}}

%e    25: {{2},{2}}

%e    28: {{},{},{1,1}}

%e    36: {{},{},{1},{1}}

%e    46: {{},{2,2}}

%e    50: {{},{2},{2}}

%e    56: {{},{},{},{1,1}}

%e    72: {{},{},{},{1},{1}}

%e    92: {{},{},{2,2}}

%e    97: {{3,3}}

%e   100: {{},{},{2},{2}}

%e   112: {{},{},{},{},{1,1}}

%e   121: {{3},{3}}

%e   144: {{},{},{},{},{1},{1}}

%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   184: {{},{},{},{2,2}}

%e   185: {{2},{1,1,2}}

%e   194: {{},{3,3}}

%e   195: {{1},{2},{1,2}}

%e   200: {{},{},{},{2},{2}}

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

%t Select[Range[100],Or[#==1,SameQ[##,2]&@@Last/@FactorInteger[Times@@primeMS[#]]]&]

%Y Cf. A003963, A005117, A005176, A062503, A064573, A072774, A295193, A302505, A319878, A319899, A320325, A322526, A322527, A322530.

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 17 2018