A318401 - OEIS

%I #12 Dec 16 2018 17:58:15

%S 1,2,3,7,13,15,19,35,37,53,61,69,89,91,95,113,131,141,143,145,151,161,

%T 165,223,247,251,265,281,299,309,311,329,355,359,377,385,407,427,437,

%U 463,503,591,593,611,655,659,667,671,689,703,719,721,759,791,827,851

%N Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices span an initial interval of positive integers.

%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 strict antichains of multisets spanning an initial interval of positive integers.

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

%e    1: {}

%e    2: {{}}

%e    3: {{1}}

%e    7: {{1,1}}

%e   13: {{1,2}}

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

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

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

%e   37: {{1,1,2}}

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

%e   61: {{1,2,2}}

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

%e   89: {{1,1,1,2}}

%e   91: {{1,1},{1,2}}

%e   95: {{2},{1,1,1}}

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

%t normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];

%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];

%t Select[Range[200],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],stableQ[primeMS[#],Divisible]]&]

%Y Cf. A003963, A006126, A055932, A056239, A112798, A285572, A290103, A293993, A302242, A304713, A316476, A319496, A319721, A319837, A320275, A320456.

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 16 2018