cp's OEIS Frontend

A319496 Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices are connected and span an initial interval of positive integers.

%I A319496 #14 Dec 16 2018 17:58:22
%S A319496 2,3,7,13,19,37,53,61,89,91,113,131,151,223,247,251,281,299,311,359,
%T A319496 377,427,463,503,593,611,659,689,703,719,791,827,851,863,923,953,1069,
%U A319496 1073,1159,1163,1291,1321,1339,1363,1511,1619,1703,1733,1739,1757,1769
%N A319496 Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices are connected and span an initial interval of positive integers.
%C A319496 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 connected strict antichains of multisets spanning an initial interval of positive integers.
%e A319496 The sequence of multisystems whose MM-numbers belong to the sequence begins:
%e A319496     2: {{}}
%e A319496     3: {{1}}
%e A319496     7: {{1,1}}
%e A319496    13: {{1,2}}
%e A319496    19: {{1,1,1}}
%e A319496    37: {{1,1,2}}
%e A319496    53: {{1,1,1,1}}
%e A319496    61: {{1,2,2}}
%e A319496    89: {{1,1,1,2}}
%e A319496    91: {{1,1},{1,2}}
%e A319496   113: {{1,2,3}}
%e A319496   131: {{1,1,1,1,1}}
%e A319496   151: {{1,1,2,2}}
%e A319496   223: {{1,1,1,1,2}}
%e A319496   247: {{1,2},{1,1,1}}
%e A319496   251: {{1,2,2,2}}
%e A319496   281: {{1,1,2,3}}
%e A319496   299: {{1,2},{2,2}}
%t A319496 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A319496 normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
%t A319496 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 A319496 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A319496 Select[Range[200],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],stableQ[primeMS[#],Divisible],Length[zsm[primeMS[#]]]==1]&]
%Y A319496 Cf. A003963, A006126, A055932, A056239, A112798, A285573, A286520, A293994, A302242, A318401, A319719, A319837, A320275, A320456, A320532.
%K A319496 nonn
%O A319496 1,1
%A A319496 _Gus Wiseman_, Dec 16 2018