cp's OEIS Frontend

A319837 Numbers whose distinct prime indices are pairwise indivisible and whose own prime indices span an initial interval of positive integers.

%I A319837 #25 Dec 16 2018 17:58:29
%S A319837 1,2,3,4,7,8,9,13,15,16,19,27,32,35,37,45,49,53,61,64,69,75,81,89,91,
%T A319837 95,113,128,131,135,141,143,145,151,161,165,169,175,207,223,225,243,
%U A319837 245,247,251,256,265,281,299,309,311,329,343,355,359,361,375,377,385
%N A319837 Numbers whose distinct prime indices are pairwise indivisible and whose own prime indices span an initial interval of positive integers.
%C A319837 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 not necessarily strict antichains of multisets spanning an initial interval of positive integers.
%e A319837 The sequence of multisystems whose MM-numbers belong to the sequence begins:
%e A319837    1: {}
%e A319837    2: {{}}
%e A319837    3: {{1}}
%e A319837    4: {{},{}}
%e A319837    7: {{1,1}}
%e A319837    8: {{},{},{}}
%e A319837    9: {{1},{1}}
%e A319837   13: {{1,2}}
%e A319837   15: {{1},{2}}
%e A319837   16: {{},{},{},{}}
%e A319837   19: {{1,1,1}}
%e A319837   27: {{1},{1},{1}}
%e A319837   32: {{},{},{},{},{}}
%e A319837   35: {{2},{1,1}}
%e A319837   37: {{1,1,2}}
%e A319837   45: {{1},{1},{2}}
%e A319837   49: {{1,1},{1,1}}
%e A319837   53: {{1,1,1,1}}
%e A319837   61: {{1,2,2}}
%e A319837   64: {{},{},{},{},{},{}}
%e A319837   69: {{1},{2,2}}
%e A319837   75: {{1},{2},{2}}
%e A319837   81: {{1},{1},{1},{1}}
%e A319837   89: {{1,1,1,2}}
%e A319837   91: {{1,1},{1,2}}
%e A319837   95: {{2},{1,1,1}}
%t A319837 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A319837 normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
%t A319837 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A319837 Select[Range[200],And[normQ[primeMS/@primeMS[#]],stableQ[primeMS[#],Divisible]]&]
%Y A319837 Cf. A003963, A006126, A055932, A056239, A112798, A285572, A290103, A293993, A302242, A304713, A316476, A318401, A319721, A320275, A320456.
%K A319837 nonn
%O A319837 1,2
%A A319837 _Gus Wiseman_, Dec 16 2018