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

1, 3, 7, 9, 13, 15, 19, 21, 27, 35, 37, 39, 45, 49, 53, 57, 61, 63, 65, 69, 75, 81, 89, 91, 95, 105, 111, 113, 117, 131, 133, 135, 141, 143, 145, 147, 151, 159, 161, 165, 169, 171, 175, 183, 185, 189, 195, 207, 223, 225, 243, 245, 247, 251, 259, 265, 267, 273

Offset: 1

Author

Gus Wiseman, Oct 18 2018

Keywords

• nonn

Comments

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}}.

Examples

			The sequence of terms together with their multiset multisystems begins:
    1: {}
    3: {{1}}
    7: {{1,1}}
    9: {{1},{1}}
   13: {{1,2}}
   15: {{1},{2}}
   19: {{1,1,1}}
   21: {{1},{1,1}}
   27: {{1},{1},{1}}
   35: {{2},{1,1}}
   37: {{1,1,2}}
   39: {{1},{1,2}}
   45: {{1},{1},{2}}
   49: {{1,1},{1,1}}
   53: {{1,1,1,1}}
   57: {{1},{1,1,1}}
   61: {{1,2,2}}
   63: {{1},{1},{1,1}}
   65: {{2},{1,2}}
   69: {{1},{2,2}}
   75: {{1},{2},{2}}
   81: {{1},{1},{1},{1}}
   89: {{1,1,1,2}}
   91: {{1,1},{1,2}}
   95: {{2},{1,1,1}}
  105: {{1},{2},{1,1}}
  111: {{1},{1,1,2}}
  113: {{1,2,3}}
  117: {{1},{1},{1,2}}
  131: {{1,1,1,1,1}}
  133: {{1,1},{1,1,1}}
  135: {{1},{1},{1},{2}}
		

Crossrefs

Odd terms of A320456.

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

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[1,100,2],normQ[primeMS/@primeMS[#]]&]