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Odd numbers whose multiset multisystem is a multiset partition spanning an initial interval of positive integers (odd = no empty sets).
2

%I #4 Oct 19 2018 09:47:25

%S 1,3,7,9,13,15,19,21,27,35,37,39,45,49,53,57,61,63,65,69,75,81,89,91,

%T 95,105,111,113,117,131,133,135,141,143,145,147,151,159,161,165,169,

%U 171,175,183,185,189,195,207,223,225,243,245,247,251,259,265,267,273

%N Odd numbers whose multiset multisystem is a multiset partition spanning an initial interval of positive integers (odd = no empty sets).

%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 3: {{1}}

%e 7: {{1,1}}

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

%e 13: {{1,2}}

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

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

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

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

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

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

%e 39: {{1},{1,2}}

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

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

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

%e 57: {{1},{1,1,1}}

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

%e 63: {{1},{1},{1,1}}

%e 65: {{2},{1,2}}

%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}}

%e 105: {{1},{2},{1,1}}

%e 111: {{1},{1,1,2}}

%e 113: {{1,2,3}}

%e 117: {{1},{1},{1,2}}

%e 131: {{1,1,1,1,1}}

%e 133: {{1,1},{1,1,1}}

%e 135: {{1},{1},{1},{2}}

%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[1,100,2],normQ[primeMS/@primeMS[#]]&]

%Y Odd terms of A320456.

%Y Cf. A003963, A055932, A056239, A112798, A255906, A302242, A305052, A320532, A320629.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 18 2018