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

1,2

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}}.

Links

Table of n, a(n) for n=1..58.

Example

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}}

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[#]]&]

Crossrefs

Odd terms of A320456.

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

Sequence in context: A086515 A132222 A340933 * A330122 A330097 A330107

Adjacent sequences: A320631 A320632 A320633 * A320635 A320636 A320637

Keyword

nonn

Author

Gus Wiseman, Oct 18 2018

Status

approved