A319878 - OEIS

%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