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Numbers whose distinct prime indices are pairwise indivisible and whose own prime indices span an initial interval of positive integers.
6

%I #25 Dec 16 2018 17:58:29

%S 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 95,113,128,131,135,141,143,145,151,161,165,169,175,207,223,225,243,

%U 245,247,251,256,265,281,299,309,311,329,343,355,359,361,375,377,385

%N Numbers whose distinct prime indices are 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 not necessarily 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 4: {{},{}}

%e 7: {{1,1}}

%e 8: {{},{},{}}

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

%e 13: {{1,2}}

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

%e 16: {{},{},{},{}}

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

%e 27: {{1},{1},{1}}

%e 32: {{},{},{},{},{}}

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

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

%e 45: {{1},{1},{2}}

%e 49: {{1,1},{1,1}}

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

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

%e 64: {{},{},{},{},{},{}}

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

%e 75: {{1},{2},{2}}

%e 81: {{1},{1},{1},{1}}

%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[normQ[primeMS/@primeMS[#]],stableQ[primeMS[#],Divisible]]&]

%Y Cf. A003963, A006126, A055932, A056239, A112798, A285572, A290103, A293993, A302242, A304713, A316476, A318401, A319721, A320275, A320456.

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 16 2018