A096827 -id:A096827 - cp's OEIS frontend

A173883 a(n) = number of iterations in the sequence of classes of prime numbers for prime(n).

0, 0, 0, 0, 0, 1, 1, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 6, 5, 4, 4, 4, 4, 4, 6, 4, 5, 4, 5, 4, 5, 5, 4, 4, 4, 4, 5, 6, 5, 4, 6, 4, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8

Offset: 2

Artur Jasinski, Mar 01 2010

nonn

Alexander S. Karpenko, Lukasiewicz's Logics and Prime Numbers, Luniver Press, Beckington, 2006, pp. 98-102.

Cf. A058340, A096827, A175177, A175178, A175179.

Edited, corrected and extended by Arkadiusz Wesolowski, Jan 19 2013

A321185 Number of integer partitions of n that are the vertex-degrees of some strict antichain of sets with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 5, 5, 9, 11, 17, 20

Offset: 0

Gus Wiseman, Oct 29 2018

A strict antichain is a finite set of finite nonempty sets, none of which is a subset of any other.

			The a(2) = 1 through a(9) = 11 partitions:
  (11)  (111)  (211)   (2111)   (222)     (2221)     (2222)      (3222)
               (1111)  (11111)  (2211)    (22111)    (3221)      (22221)
                                (3111)    (31111)    (22211)     (32211)
                                (21111)   (211111)   (32111)     (33111)
                                (111111)  (1111111)  (41111)     (222111)
                                                     (221111)    (321111)
                                                     (311111)    (411111)
                                                     (2111111)   (2211111)
                                                     (11111111)  (3111111)
                                                                 (21111111)
                                                                 (111111111)
The a(8) = 9 integer partitions together with a realizing strict antichain for each (the parts of the partition count the appearances of each vertex in the antichain):
     (41111): {{1,2},{1,3},{1,4},{1,5}}
      (3221): {{1,2},{1,3},{1,4},{2,3}}
     (32111): {{1,3},{1,2,4},{1,2,5}}
    (311111): {{1,2},{1,3},{1,4,5,6}}
      (2222): {{1,2},{1,3,4},{2,3,4}}
     (22211): {{1,2,3,4},{1,2,3,5}}
    (221111): {{1,2,3},{1,2,4,5,6}}
   (2111111): {{1,2},{1,3,4,5,6,7}}
  (11111111): {{1,2,3,4,5,6,7,8}}

Cf. A000070, A000569, A006126, A096827, A209816, A283877, A293606, A293993, A306005, A318361, A319721, A321176, A321184.

submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
stableQ[u_]:=Apply[And,Outer[#1==#2||!submultisetQ[#1,#2]&&!submultisetQ[#2,#1]&,u,u,1],{0,1}];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
anti[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,UnsameQ@@#,Min@@Length/@#>1,stableQ[#]]&];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[n],anti[#]!={}&]],{n,8}]

cp's OEIS Frontend

A173883 a(n) = number of iterations in the sequence of classes of prime numbers for prime(n).

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A321185 Number of integer partitions of n that are the vertex-degrees of some strict antichain of sets with no singletons.

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