A320456 - OEIS

%I #5 Oct 15 2018 22:20:52

%S 1,2,3,4,6,7,8,9,12,13,14,15,16,18,19,21,24,26,27,28,30,32,35,36,37,

%T 38,39,42,45,48,49,52,53,54,56,57,60,61,63,64,65,69,70,72,74,75,76,78,

%U 81,84,89,90,91,95,96,98,104,105,106,108,111,112,113,114,117

%N Numbers whose multiset multisystem spans 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 n-th multiset multisystem 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 78th multiset multisystem is {{},{1},{1,2}}.

%e The sequence of terms together with their multiset multisystems begins:

%e    1: {}

%e    2: {{}}

%e    3: {{1}}

%e    4: {{},{}}

%e    6: {{},{1}}

%e    7: {{1,1}}

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

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

%e   12: {{},{},{1}}

%e   13: {{1,2}}

%e   14: {{},{1,1}}

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

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

%e   18: {{},{1},{1}}

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

%e   21: {{1},{1,1}}

%e   24: {{},{},{},{1}}

%e   26: {{},{1,2}}

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

%e   28: {{},{},{1,1}}

%e   30: {{},{1},{2}}

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

%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 Select[Range[100],normQ[primeMS/@primeMS[#]]&]

%Y Cf. A001222, A003963, A034691, A034729, A055932, A056239, A112798, A255906, A290103, A302242, A305052.

%Y Cf. A320458, A320459, A320461, A320462, A320463, A320464, A320532, A320533.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 13 2018