A305001 -id:A305001 - cp's OEIS frontend

A006602 a(n) is the number of hierarchical models on n unlabeled factors or variables with linear terms forced.

2, 1, 2, 5, 20, 180, 16143, 489996795, 1392195548399980210, 789204635842035039135545297410259322

Offset: 0

Also number of pure (= irreducible) group-testing histories of n items - A. Boneh, Mar 31 2000
Also number of antichain covers of an unlabeled n-set, so a(n) equals first differences of A003182. - Vladeta Jovovic, Goran Kilibarda, Aug 18 2000
Also number of inequivalent (under permutation of variables) nondegenerate monotone Boolean functions of n variables. We say h and g (functions of n variables) are equivalent if there exists a permutation p of S_n such that hp=g. E.g., a(3)=5 because xyz, xy+xz+yz, x+yz+xyz, xy+xz+xyz, x+y+z+xy+xz+yz+xyz are 5 inequivalent nondegenerate monotone Boolean functions that generate (by permutation of variables) the other 4. For example, y+xz+xyz can be obtained from x+yz+xyz by exchanging x and y. - Alan Veliz-Cuba (alanavc(AT)vt.edu), Jun 16 2006
The non-spanning/covering case is A003182. The labeled case is A006126. - Gus Wiseman, Feb 20 2019

			From _Gus Wiseman_, Feb 20 2019: (Start)
Non-isomorphic representatives of the a(0) = 2 through a(4) = 20 antichains:
  {}    {{1}}  {{12}}    {{123}}         {{1234}}
  {{}}         {{1}{2}}  {{1}{23}}       {{1}{234}}
                         {{13}{23}}      {{12}{34}}
                         {{1}{2}{3}}     {{14}{234}}
                         {{12}{13}{23}}  {{1}{2}{34}}
                                         {{134}{234}}
                                         {{1}{24}{34}}
                                         {{1}{2}{3}{4}}
                                         {{13}{24}{34}}
                                         {{14}{24}{34}}
                                         {{13}{14}{234}}
                                         {{12}{134}{234}}
                                         {{1}{23}{24}{34}}
                                         {{124}{134}{234}}
                                         {{12}{13}{24}{34}}
                                         {{14}{23}{24}{34}}
                                         {{12}{13}{14}{234}}
                                         {{123}{124}{134}{234}}
                                         {{13}{14}{23}{24}{34}}
                                         {{12}{13}{14}{23}{24}{34}}
(End)

Y. M. M. Bishop, S. E. Fienberg and P. W. Holland, Discrete Multivariate Analysis. MIT Press, 1975, p. 34. [In part (e), the Hierarchy Principle for log-linear models is defined. It essentially says that if a higher-order parameter term is included in the log-linear model, then all the lower-order parameter terms should also be included. - Petros Hadjicostas, Apr 10 2020]
V. Jovovic and G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.
A. A. Mcintosh, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See p. 14.
V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11(4) (1999), 127-138 (translated in Discrete Mathematics and Applications, 9(6) (1999), 593-605).
C. Lienkaemper, When do neural codes come from convex or good covers?, 2015.
C. L. Mallows, Emails to N. J. A. Sloane, Jun-Jul 1991
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A000372, A003182, A006126 (labeled case), A007411, A014466, A261005, A293993, A304997, A304998, A304999, A305001, A305855, A306505, A320449, A321679.

a(n) = A007411(n) + 1.

First differences of A003182. - Gus Wiseman, Feb 23 2019

a(6) from A. Boneh, 32 Hantkeh St., Haifa 34608, Israel, Mar 31 2000
Entry revised by N. J. A. Sloane, Jul 23 2006
a(7) from A007411 and A003182. - N. J. A. Sloane, Aug 13 2015
Named edited by Petros Hadjicostas, Apr 08 2020
a(8) from A003182. - Bartlomiej Pawelski, Nov 27 2022
a(9) from A007411. - Dmitry I. Ignatov, Nov 27 2023

A305000 Number of labeled antichains of finite sets spanning some subset of {1,...,n} with singleton edges allowed.

Original entry on oeis.org

1, 2, 8, 72, 1824, 220608, 498243968, 309072306743552, 14369391925598802012151296, 146629927766168786239127150948525247729660416

Offset: 0

Gus Wiseman, May 23 2018

Only the non-singleton edges are required to form an antichain.

Number of non-degenerate unate Boolean functions of n or fewer variables. - Aniruddha Biswas, May 11 2024

			The a(2) = 8 antichains:
  {}
  {{1}}
  {{2}}
  {{1,2}}
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}
  {{1},{2},{1,2}}

Aniruddha Biswas and Palash Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See pp. 4, 12.
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A006126, A261005, A304996, A304997, A304998, A304999, A305000, A305001.
Cf. A000372, A003182, A014466, A293606, A293993, A306505, A320449, A321679.
Cf. A245079.

Binomial transform of A304999.

Inverse binomial transform of A245079. - Aniruddha Biswas, May 11 2024

a(5)-a(8) from Gus Wiseman, May 31 2018

a(9) from Aniruddha Biswas, May 11 2024

A305052 z-density of the integer partition with Heinz number n. Clutter density of the n-th multiset multisystem (A302242).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 24 2018

sign

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The z-density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(lcm(S)) where omega = A001221 is number of distinct prime factors.
First nonnegative entry after a(1) = 0 is a(169) = 0.

			The 1105th multiset multisystem is {{2},{1,2},{4}} with clutter density -2, so a(1105) = -2.
The 5429th multiset multisystem is {{1,2,2},{1,1,1,2}} with clutter density 0, so a(5429) = 0.
The 11837th multiset multisystem is {{1,1},{1,1,1},{1,1,1,2}} with clutter density -1, so a(11837) = -1.
The 42601th multiset multisystem is {{1,2},{1,3},{1,2,3}} with clutter density 1, so a(42601) = 1.

Cf. A001221, A030019, A048143, A056239, A112798, A285572, A286518, A286520, A290103, A302242, A303837, A304118, A304714, A304716, A304911, A305001.

zens[n_]:=If[n==1,0,Total@Cases[FactorInteger[n],{p_,k_}:>k*(PrimeNu[PrimePi[p]]-1)]-PrimeNu[LCM@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]];
Array[zens,100]

A304998 Number of unlabeled antichains of finite sets spanning n vertices without singletons.

Original entry on oeis.org

1, 0, 1, 3, 15, 160, 15963, 489980652

Offset: 0

Gus Wiseman, May 23 2018

nonn

			Non-isomorphic representatives of the a(4) = 15 antichains:
  {{1,2,3,4}}
  {{1,2},{3,4}}
  {{1,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4},{3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Cf. A006126, A261005, A304996, A304997, A304998, A304999, A305000, A305001.

a(n > 0) = A261005(n) - A261005(n - 1).

A305149 Number of factorizations of n whose distinct factors are pairwise indivisible and greater than 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 26 2018

nonn

			The a(60) = 8 factorizations are (2*2*3*5), (2*2*15), (3*4*5), (3*20), (4*15), (5*12), (6*10), (60). Missing from this list are (2*3*10), (2*5*6), (2*30).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A001055, A001970, A007716, A034444, A045778, A259936, A281116, A285572, A302242, A303386, A303431, A305001, A305148, A305150.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Select[Tuples[Union[#],2],UnsameQ@@#&&Divisible@@#&]=={}&]],{n,100}]

pairwise_indivisible(v) = { for(i=1,#v,for(j=i+1,#v,if(!(v[j]%v[i]),return(0)))); (1); };
A305149(n, m=n, facs=List([])) = if(1==n, pairwise_indivisible(Set(facs)), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A305149(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Oct 08 2018

More terms from Antti Karttunen, Oct 08 2018

A326358 Number of maximal antichains of subsets of {1..n}.

Original entry on oeis.org

1, 2, 3, 7, 29, 376, 31746, 123805914

Offset: 0

Gus Wiseman, Jul 01 2019

A set system (set of sets) is an antichain if no element is a subset of any other.

			The a(0) = 1 through a(3) = 7 maximal antichains:
  {}  {}   {}      {}
      {1}  {12}    {123}
           {1}{2}  {1}{23}
                   {2}{13}
                   {3}{12}
                   {1}{2}{3}
                   {12}{13}{23}

Denis Bouyssou, Thierry Marchant, and Marc Pirlot, ELECTRE TRI-nB, pseudo-disjunctive: axiomatic and combinatorial results, arXiv:2410.18443 [cs.DM], 2024. See p. 14.
Dmitry I. Ignatov, A Note on the Number of (Maximal) Antichains in the Lattice of Set Partitions. In: Ojeda-Aciego, M., Sauerwald, K., Jäschke, R. (eds) Graph-Based Representation and Reasoning. ICCS 2023. Lecture Notes in Computer Science(). Springer, Cham.
Dmitry I. Ignatov, On the Number of Maximal Antichains in Boolean Lattices for n up to 7. Lobachevskii J. Math., 44 (2023), 137-146.

Antichains of sets are A000372.
Minimal covering antichains are A046165.
Maximal intersecting antichains are A007363.
Maximal antichains of nonempty sets are A326359.
Cf. A003182, A006126, A006602, A014466, A058891, A261005, A305000, A305001, A305844, A326360, A326361, A326362, A326363.

LoadPackage("grape");
      maxachP:=function(n) local g,G;
       g:=Graph(Group(()), Combinations([1..n]), function(x, g) return x; end,
          function(x, y) return not IsSubset(x, y) and not IsSubset(y, x); end, true);
       G:=AutGroupGraph(g);
       return Sum(CompleteSubgraphs(NewGroupGraph(G, g), -1, 2),
              function(c) return Length(Orbit(G, c, OnSets)); end);
     end;
       List([0..7],maxachP); # Mamuka Jibladze, Jan 26 2021

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[stableSets[Subsets[Range[n]],SubsetQ]]],{n,0,5}]
(* alternatively *)
maxachP[n_]:=FindIndependentVertexSet[
  Flatten[Map[Function[s, Map[# \[DirectedEdge] s &, Most[Subsets[s]]]],
    Subsets[Range[n]]]], Infinity, All];
Table[Length[maxachP[n]],{n,0,6}] (* Mamuka Jibladze, Jan 25 2021 *)

For n > 0, a(n) = A326359(n) + 1.

a(6)-a(7) from Mamuka Jibladze, Jan 26 2021

A304997 Number of unlabeled antichains of finite sets spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 4, 18, 142, 3100, 823042

Offset: 0

Gus Wiseman, May 23 2018

			Non-isomorphic representatives of the a(3) = 18 antichains:
{{1,2,3}}
{{3},{1,2}}
{{3},{1,2,3}}
{{1,3},{2,3}}
{{1},{2},{3}}
{{2},{3},{1,3}}
{{2},{3},{1,2,3}}
{{3},{1,2},{2,3}}
{{3},{1,3},{2,3}}
{{1,2},{1,3},{2,3}}
{{1},{2},{3},{2,3}}
{{1},{2},{3},{1,2,3}}
{{2},{3},{1,2},{1,3}}
{{2},{3},{1,3},{2,3}}
{{3},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,3},{2,3}}
{{2},{3},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,2},{1,3},{2,3}}

Cf. A006126, A261005, A304996, A304997, A304998, A304999, A305000, A305001.

A305150 Number of factorizations of n into distinct, pairwise indivisible factors greater than 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 2, 2, 1, 2, 2, 3, 1, 5, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 6, 1, 2, 2, 1, 2, 5, 1, 2, 2, 5, 1, 3, 1, 2, 2, 2, 2, 5, 1, 3, 1, 2, 1, 6, 2, 2, 2, 3, 1, 6, 2, 2, 2, 2, 2, 4, 1, 2, 2, 2, 1, 5, 1, 3, 5

Offset: 1

Gus Wiseman, May 26 2018

nonn

			The a(60) = 6 factorizations are (3 * 4 * 5), (3 * 20), (4 * 15), (5 * 12), (6 * 10), (60). Missing from this list are (2 * 3 * 10), (2 * 5 * 6), (2 * 30).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A001055, A001970, A007716, A045778, A162247, A259936, A275024, A285572, A281113, A281116, A303386, A303431, A305001, A305148, A305149, A305253.

facs[n_] := If[n <= 1, {{}}, Join@@Table[Map[Prepend[#, d] &, Select[facs[n/d], Min@@ # >= d &]], {d, Rest[Divisors[n]]}]]; Table[Length[Select[facs[n], UnsameQ@@ # && Select[Tuples[Union[#], 2], UnsameQ@@ # && Divisible@@ # &] == {} &]], {n, 100}]

A305150(n, m=n, facs=List([])) = if(1==n, 1, my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m)&&factorback(apply(x -> (x%d),Vec(facs))), newfacs = List(facs); listput(newfacs,d); s += A305150(n/d, d-1, newfacs))); (s)); \\ Antti Karttunen, Dec 06 2018

a(n) <= A045778(n) <= A001055(n). - Antti Karttunen, Dec 06 2018

More terms from Antti Karttunen, Dec 06 2018

A306505 Number of non-isomorphic antichains of nonempty subsets of {1,...,n}.

Original entry on oeis.org

1, 2, 4, 9, 29, 209, 16352, 490013147, 1392195548889993357, 789204635842035040527740846300252679

Offset: 0

Gus Wiseman, Feb 20 2019

The spanning case is A006602 or A261005. The labeled case is A014466.
From Petros Hadjicostas, Apr 22 2020: (Start)
a(n) is the number of "types" of log-linear hierarchical models on n factors in the sense of Colin Mallows (see the emails to N. J. A. Sloane).
Two hierarchical models on n factors belong to the same "type" iff one can obtained from the other by a permutation of the factors.
The total number of hierarchical log-linear models on n factors (in all "types") is given by A014466(n) = A000372(n) - 1.
The name of a hierarchical log-linear model on factors is based on the collection of maximal interaction terms, which must be an antichain (by the definition of maximality).
In his example on p. 1, Colin Mallows groups the A014466(3) = 19 hierarchical log-linear models on n = 3 factors x, y, z into a(3) = 9 types. See my example below for more details. (End)
First differs from A348260(n + 1) - 1 at a(5) = 209, A348260(6) - 1 = 232. - Gus Wiseman, Nov 28 2021

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 9 antichains:
  {}  {}     {}         {}
      {{1}}  {{1}}      {{1}}
             {{1,2}}    {{1,2}}
             {{1},{2}}  {{1},{2}}
                        {{1,2,3}}
                        {{1},{2,3}}
                        {{1},{2},{3}}
                        {{1,3},{2,3}}
                        {{1,2},{1,3},{2,3}}
From _Petros Hadjicostas_, Apr 23 2020: (Start)
We expand _Colin Mallows_'s example from p. 1 of his list of 1991 emails. For n = 3, we have the following a(3) = 9 "types" of log-linear hierarchical models:
Type 1: ( ), Type 2: (x), (y), (z), Type 3: (x,y), (y,z), (z,x), Type 4: (x,y,z), Type 5: (xy), (yz), (zx), Type 6: (xy,z), (yz,x), (zx,y), Type 7: (xy,xz), (yx,yz), (zx,zy), Type 8: (xy,yz,zx), Type 9: (xyz).
For each model, the name only contains the maximal terms. See p. 36 in Wickramasinghe (2008) for the full description of the 19 models.
Strictly speaking, I should have used set notation (rather than parentheses) for the name of each model, but I follow the tradition of the theory of log-linear models. In addition, in an interaction term such as xy, the order of the factors is irrelevant.
Models in the same type essentially have similar statistical properties.
For example, models in Type 7 have the property that two factors are conditionally independent of one another given each level (= category) of the third factor.
Models in Type 6 are such that two factors are jointly independent from the third one. (End)

C. L. Mallows, Emails to N. J. A. Sloane, Jun-Jul 1991, p. 1.
R. I. P. Wickramasinghe, Topics in log-linear models, Master of Science thesis in Statistics, Texas Tech University, Lubbock, TX, 2008, p. 36.
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A000372, A003182, A006126, A006602, A014466, A261005, A293606, A293993, A304996, A305000, A305001, A305857, A317674, A319721, A320449, A321679.

Cf. A007363, A306007, A307249, A326358, A326359, A326360, A326363.

a(n) = A003182(n) - 1.

Partial sums of A006602 minus 1.

a(8) from A003182. - Bartlomiej Pawelski, Nov 27 2022

a(9) from A003182. - Dmitry I. Ignatov, Nov 27 2023

A304996 Number of unlabeled antichains of finite sets spanning up to n vertices with singleton edges allowed.

Original entry on oeis.org

1, 2, 6, 24, 166, 3266, 826308

Offset: 0

Gus Wiseman, May 23 2018

			Non-isomorphic representatives of the a(3) = 24 antichains:
{}
{{1}}
{{1,2}}
{{1,2,3}}
{{1},{2}}
{{2},{1,2}}
{{3},{1,2}}
{{3},{1,2,3}}
{{1,3},{2,3}}
{{1},{2},{3}}
{{1},{2},{1,2}}
{{2},{3},{1,3}}
{{2},{3},{1,2,3}}
{{3},{1,2},{2,3}}
{{3},{1,3},{2,3}}
{{1,2},{1,3},{2,3}}
{{1},{2},{3},{2,3}}
{{1},{2},{3},{1,2,3}}
{{2},{3},{1,2},{1,3}}
{{2},{3},{1,3},{2,3}}
{{3},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,3},{2,3}}
{{2},{3},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,2},{1,3},{2,3}}

Partial sums of A304997.
Cf. A006126, A261005, A304997, A304998, A304999, A305000, A305001.
Cf. A000372, A014466, A293993, A319639, A319721, A326704, A326972.

a(5)-a(6) from Andrew Howroyd, Aug 14 2019

cp's OEIS Frontend

A006602 a(n) is the number of hierarchical models on n unlabeled factors or variables with linear terms forced.

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A305000 Number of labeled antichains of finite sets spanning some subset of {1,...,n} with singleton edges allowed.

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A305052 z-density of the integer partition with Heinz number n. Clutter density of the n-th multiset multisystem (A302242).

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A304998 Number of unlabeled antichains of finite sets spanning n vertices without singletons.

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A305149 Number of factorizations of n whose distinct factors are pairwise indivisible and greater than 1.

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A326358 Number of maximal antichains of subsets of {1..n}.

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GAP

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A304997 Number of unlabeled antichains of finite sets spanning n vertices with singleton edges allowed.

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A305150 Number of factorizations of n into distinct, pairwise indivisible factors greater than 1.

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