cp's OEIS Frontend

A320634 Odd numbers whose multiset multisystem is a multiset partition spanning an initial interval of positive integers (odd = no empty sets).

%I A320634 #4 Oct 19 2018 09:47:25
%S A320634 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 A320634 95,105,111,113,117,131,133,135,141,143,145,147,151,159,161,165,169,
%U A320634 171,175,183,185,189,195,207,223,225,243,245,247,251,259,265,267,273
%N A320634 Odd numbers whose multiset multisystem is a multiset partition spanning an initial interval of positive integers (odd = no empty sets).
%C A320634 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 A320634 The sequence of terms together with their multiset multisystems begins:
%e A320634     1: {}
%e A320634     3: {{1}}
%e A320634     7: {{1,1}}
%e A320634     9: {{1},{1}}
%e A320634    13: {{1,2}}
%e A320634    15: {{1},{2}}
%e A320634    19: {{1,1,1}}
%e A320634    21: {{1},{1,1}}
%e A320634    27: {{1},{1},{1}}
%e A320634    35: {{2},{1,1}}
%e A320634    37: {{1,1,2}}
%e A320634    39: {{1},{1,2}}
%e A320634    45: {{1},{1},{2}}
%e A320634    49: {{1,1},{1,1}}
%e A320634    53: {{1,1,1,1}}
%e A320634    57: {{1},{1,1,1}}
%e A320634    61: {{1,2,2}}
%e A320634    63: {{1},{1},{1,1}}
%e A320634    65: {{2},{1,2}}
%e A320634    69: {{1},{2,2}}
%e A320634    75: {{1},{2},{2}}
%e A320634    81: {{1},{1},{1},{1}}
%e A320634    89: {{1,1,1,2}}
%e A320634    91: {{1,1},{1,2}}
%e A320634    95: {{2},{1,1,1}}
%e A320634   105: {{1},{2},{1,1}}
%e A320634   111: {{1},{1,1,2}}
%e A320634   113: {{1,2,3}}
%e A320634   117: {{1},{1},{1,2}}
%e A320634   131: {{1,1,1,1,1}}
%e A320634   133: {{1,1},{1,1,1}}
%e A320634   135: {{1},{1},{1},{2}}
%t A320634 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A320634 normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
%t A320634 Select[Range[1,100,2],normQ[primeMS/@primeMS[#]]&]
%Y A320634 Odd terms of A320456.
%Y A320634 Cf. A003963, A055932, A056239, A112798, A255906, A302242, A305052, A320532, A320629.
%K A320634 nonn
%O A320634 1,2
%A A320634 _Gus Wiseman_, Oct 18 2018