cp's OEIS Frontend

A320275 Numbers whose distinct prime indices are pairwise indivisible and whose own prime indices are connected and span an initial interval of positive integers.

%I A320275 #19 Dec 20 2018 22:44:52
%S A320275 2,3,7,9,13,19,27,37,49,53,61,81,89,91,113,131,151,169,223,243,247,
%T A320275 251,281,299,311,343,359,361,377,427,463,503,593,611,637,659,689,703,
%U A320275 719,729,791,827,851,863,923,953,1069,1073,1159,1163,1183,1291,1321,1339
%N A320275 Numbers whose distinct prime indices are pairwise indivisible and whose own prime indices are connected and span an initial interval of positive integers.
%C A320275 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 connected antichains of multisets spanning an initial interval of positive integers.
%e A320275 The sequence of multisystems whose MM-numbers belong to the sequence begins:
%e A320275     2: {{}}
%e A320275     3: {{1}}
%e A320275     7: {{1,1}}
%e A320275     9: {{1},{1}}
%e A320275    13: {{1,2}}
%e A320275    19: {{1,1,1}}
%e A320275    27: {{1},{1},{1}}
%e A320275    37: {{1,1,2}}
%e A320275    49: {{1,1},{1,1}}
%e A320275    53: {{1,1,1,1}}
%e A320275    61: {{1,2,2}}
%e A320275    81: {{1},{1},{1},{1}}
%e A320275    89: {{1,1,1,2}}
%e A320275    91: {{1,1},{1,2}}
%e A320275   113: {{1,2,3}}
%e A320275   131: {{1,1,1,1,1}}
%e A320275   151: {{1,1,2,2}}
%e A320275   169: {{1,2},{1,2}}
%t A320275 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A320275 normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
%t A320275 zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
%t A320275 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A320275 Select[Range[200],And[normQ[primeMS/@primeMS[#]],stableQ[primeMS[#],Divisible],Length[zsm[primeMS[#]]]==1]&]
%Y A320275 Cf. A003963, A006126, A055932, A056239, A112798, A285572, A286520, A290103, A293994, A302242, A316476, A319496, A319837, A320456, A320532.
%K A320275 nonn
%O A320275 1,1
%A A320275 _Gus Wiseman_, Dec 16 2018