A285572 -id:A285572 - cp's OEIS frontend

A326082 Number of maximal sets of pairwise indivisible divisors of n.

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 5, 2, 3, 3, 5, 2, 5, 2, 5, 3, 3, 2, 8, 3, 3, 4, 5, 2, 7, 2, 6, 3, 3, 3, 9, 2, 3, 3, 8, 2, 7, 2, 5, 5, 3, 2, 12, 3, 5, 3, 5, 2, 8, 3, 8, 3, 3, 2, 15, 2, 3, 5, 7, 3, 7, 2, 5, 3, 7, 2, 15, 2, 3, 5, 5, 3, 7, 2, 12, 5, 3, 2, 15, 3

Offset: 1

Gus Wiseman, Jun 05 2019

nonn

Depends only on prime signature.

The non-maximal case is A096827.

			The maximal sets of pairwise indivisible divisors of n = 1, 2, 4, 8, 12, 24, 30, 32, 36, 48, 60 are:
   1   1   1   1   1     1      1         1    1       1       1
       2   2   2   12    24     30        2    36      48      60
           4   4   2,3   2,3    5,6       4    2,3     2,3     2,15
               8   3,4   3,4    2,15      8    2,9     3,4     3,20
                   4,6   3,8    3,10      16   3,4     3,8     4,30
                         4,6    2,3,5     32   4,18    4,6     5,12
                         6,8    6,10,15        9,12    6,8     2,3,5
                         8,12                  12,18   3,16    3,4,5
                                               4,6,9   6,16    4,5,6
                                                       8,12    3,4,10
                                                       12,16   6,15,20
                                                       16,24   10,12,15
                                                               12,15,20
                                                               12,20,30
                                                               4,6,10,15

Cf. A001055, A051026, A067992, A096827, A143824, A285572, A285573, A303362, A305148, A305149, A316476, A325861, A326023, A326077.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Rest[Subsets[Divisors[n]]],stableQ[#,Divisible]&]]],{n,100}]

A328678 Number of strict, pairwise indivisible, relatively prime integer partitions of n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 1, 2, 2, 4, 3, 5, 4, 5, 7, 10, 9, 12, 11, 14, 15, 22, 20, 25, 26, 32, 33, 44, 41, 54, 49, 62, 67, 80, 80, 100, 100, 118, 121, 152, 148, 179, 178, 210, 219, 267, 259, 316, 313, 363, 380, 449, 448, 529, 532, 619, 640, 745, 749, 867, 889

Offset: 1

Gus Wiseman, Oct 30 2019

nonn

Note that pairwise indivisibility implies strictness, but we include "strict" in the name in order to more clearly distinguish it from A328676 = "Number of relatively prime integer partitions of n whose distinct parts are pairwise indivisible".

			The a(1) = 1 through a(20) = 11 partitions (A..H = 10..20) (empty columns not shown):
  1  32  43  53  54  73   65  75   76  95   87   97   98    B7   A9    B9
         52      72  532  74  543  85  B3   B4   B5   A7    D5   B8    D7
                          83  732  94  743  D2   D3   B6    765  C7    H3
                          92       A3  752  654  754  C5    873  D6    875
                                   B2       753  853  D4    954  E5    965
                                                 952  E3    972  F4    974
                                                 B32  F2    B43  G3    A73
                                                      764   B52  H2    B54
                                                      A43   D32  865   B72
                                                      7532       964   D43
                                                                 B53   D52
                                                                 7543

The Heinz numbers of these partitions are the squarefree terms of A328677.
The non-strict case is A328676.
Pairwise indivisible partitions are A303362.
Strict, relatively prime partitions are A078374.
A ranking function using binary indices is A328671.
Cf. A000837, A285572, A285573, A304713, A305148, A316476, A328171.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&GCD@@#==1&&stableQ[#,Divisible]&]],{n,30}]

Moebius transform of A303362.

A330225 Position of first appearance of n in A290103 = LCM of prime indices.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

Gus Wiseman, Mar 26 2020

nonn

Appears to be the prime numbers (A000040) with 2 replaced by 1 and 37 replaced by 35.

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 version for product instead of lcm is A318871
The version for standard compositions is A333225.
The version for binary indices is A333492.
Let q(k) be the prime indices of k:
- The product of q(k) is A003963(k).
- The sum of q(k) is A056239(k).
- The terms of q(k) are row k of A112798.
- The GCD of q(k) is A289508(k).
- The LCM of q(k) is A290103(k).
- The LCM of q(k) + 1 is A328219(k).
Cf. A000837, A074761, A074971, A076078, A285572, A289509, A290104, A319333, A324837, A328451, A331579, A333226.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[If[n==1,1,LCM@@primeMS[n]],{n,100}];
Table[Position[q,i][[1,1]],{i,First[Split[Union[q],#1+1==#2&]]}]

A305254 Number of factorizations f of n into factors greater than 1 such that the graph of f is a forest.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 8, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 11, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 12, 5, 2, 1, 11, 2

Offset: 1

Gus Wiseman, May 28 2018

nonn

Given a factorization f consisting of positive integers greater than one, let G(F) be a multigraph with one vertex for each factor and n edges between any two vertices with n common divisors greater than 1. For example, G(6,14,15,35) is a 4-cycle; G(6,12) is a 2-cycle because 6 and 12 have multiple common divisors. This sequence counts factorizations f such that G(f) is a forest, meaning it has no cycles. [Comment edited by Robert Munafo, Mar 24 2024]

			The a(72) = 14 factorizations:
     (72)
    (2*36)     (3*24)    (4*18)    (8*9)
   (2*2*18)   (2*3*12)   (2*4*9)  (3*3*8) (3*4*6)
   (2*2*2*9)  (2*2*3*6) (2*3*3*4)
  (2*2*2*3*3)
not counted: (2*6*6) because 6 and 6 share multiple divisors; likewise (6*12) because 6 and 12 share multiple divisors.

Cf. A001970, A048143, A281116, A285572, A286518, A286520, A303386, A304714, A304716, A305149, A305193, A305253.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
Table[Length[Select[facs[n],Function[f,And@@(zensity[Select[f,Function[x,Divisible[#,x]]]]==-1&/@zsm[f])]]],{n,200}]

Extensive clarification by Robert Munafo, Mar 22 2024

A321678 Number of non-isomorphic weight-n strict antichains of sets with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 11, 13, 39, 67, 174

Offset: 0

Gus Wiseman, Nov 16 2018

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(2) = 1 through a(6) = 11 antichains:
  {{1,2}}  {{1,2,3}}  {{1,2,3,4}}    {{1,2,3,4,5}}    {{1,2,3,4,5,6}}
                      {{1,2},{3,4}}  {{1,2},{3,4,5}}  {{1,2},{3,4,5,6}}
                      {{1,3},{2,3}}  {{1,4},{2,3,4}}  {{1,2,3},{4,5,6}}
                                                      {{1,2,5},{3,4,5}}
                                                      {{1,3,4},{2,3,4}}
                                                      {{1,5},{2,3,4,5}}
                                                      {{1,2},{1,3},{2,3}}
                                                      {{1,2},{3,4},{5,6}}
                                                      {{1,2},{3,5},{4,5}}
                                                      {{1,3},{2,4},{3,4}}
                                                      {{1,4},{2,4},{3,4}}

Cf. A006126, A049311, A096827, A285572, A293993, A293994, A318099, A319719, A319721, A320799, A321679.

cp's OEIS Frontend

A326082 Number of maximal sets of pairwise indivisible divisors of n.

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A328678 Number of strict, pairwise indivisible, relatively prime integer partitions of n.

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A330225 Position of first appearance of n in A290103 = LCM of prime indices.

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A305254 Number of factorizations f of n into factors greater than 1 such that the graph of f is a forest.

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A321678 Number of non-isomorphic weight-n strict antichains of sets with no singletons.

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