cp's OEIS Frontend

A318401 Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices span an initial interval of positive integers.

%I A318401 #12 Dec 16 2018 17:58:15
%S A318401 1,2,3,7,13,15,19,35,37,53,61,69,89,91,95,113,131,141,143,145,151,161,
%T A318401 165,223,247,251,265,281,299,309,311,329,355,359,377,385,407,427,437,
%U A318401 463,503,591,593,611,655,659,667,671,689,703,719,721,759,791,827,851
%N A318401 Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices span an initial interval of positive integers.
%C A318401 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 A318401 The sequence of multisystems whose MM-numbers belong to the sequence begins:
%e A318401    1: {}
%e A318401    2: {{}}
%e A318401    3: {{1}}
%e A318401    7: {{1,1}}
%e A318401   13: {{1,2}}
%e A318401   15: {{1},{2}}
%e A318401   19: {{1,1,1}}
%e A318401   35: {{2},{1,1}}
%e A318401   37: {{1,1,2}}
%e A318401   53: {{1,1,1,1}}
%e A318401   61: {{1,2,2}}
%e A318401   69: {{1},{2,2}}
%e A318401   89: {{1,1,1,2}}
%e A318401   91: {{1,1},{1,2}}
%e A318401   95: {{2},{1,1,1}}
%t A318401 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A318401 normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
%t A318401 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A318401 Select[Range[200],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],stableQ[primeMS[#],Divisible]]&]
%Y A318401 Cf. A003963, A006126, A055932, A056239, A112798, A285572, A290103, A293993, A302242, A304713, A316476, A319496, A319721, A319837, A320275, A320456.
%K A318401 nonn
%O A318401 1,2
%A A318401 _Gus Wiseman_, Dec 16 2018