A305843 -id:A305843 - cp's OEIS frontend

A337667 Number of compositions of n where any two parts have a common divisor > 1.

1, 0, 1, 1, 2, 1, 5, 1, 8, 4, 17, 1, 38, 1, 65, 19, 128, 1, 284, 1, 518, 67, 1025, 1, 2168, 16, 4097, 256, 8198, 1, 16907, 7, 32768, 1027, 65537, 79, 133088, 19, 262145, 4099, 524408, 25, 1056731, 51, 2097158, 16636, 4194317, 79, 8421248, 196, 16777712

Offset: 0

Gus Wiseman, Oct 05 2020

nonn

First differs from A178472 at a(31) = 7, a(31) = 1.

			The a(2) = 1 through a(10) = 17 compositions (A = 10):
   2   3   4    5   6     7   8      9     A
           22       24        26     36    28
                    33        44     63    46
                    42        62     333   55
                    222       224          64
                              242          82
                              422          226
                              2222         244
                                           262
                                           424
                                           442
                                           622
                                           2224
                                           2242
                                           2422
                                           4222
                                           22222

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..290

A101268 = 1 + A337462 is the pairwise coprime version.
A328673 = A200976 + 1 is the unordered version.
A337604 counts these compositions of length 3.
A337666 ranks these compositions.
A337694 gives Heinz numbers of the unordered version.
A337983 is the strict case.
A051185 counts intersecting set-systems, with spanning case A305843.
A318717 is the unordered strict case.
A319786 is the version for factorizations, with strict case A318749.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A051424, A082024, A178472, A284825, A319752, A327039, A337599, A337605, A337665.

stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],stabQ[#,CoprimeQ]&]],{n,0,15}]

A326910 BII-numbers of pairwise intersecting set-systems.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 16, 17, 20, 21, 24, 32, 34, 36, 38, 40, 48, 52, 56, 64, 65, 66, 68, 69, 70, 72, 80, 81, 84, 85, 88, 96, 98, 100, 102, 104, 112, 116, 120, 128, 256, 257, 260, 261, 272, 273, 276, 277, 320, 321, 324, 325, 336, 337, 340, 341, 384, 512, 514

Offset: 1

Gus Wiseman, Aug 04 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

			The sequence of all pairwise intersecting set-systems together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  20: {{1,2},{1,3}}
  21: {{1},{1,2},{1,3}}
  24: {{3},{1,3}}
  32: {{2,3}}
  34: {{2},{2,3}}
  36: {{1,2},{2,3}}
  38: {{2},{1,2},{2,3}}
  40: {{3},{2,3}}
  48: {{1,3},{2,3}}
  52: {{1,2},{1,3},{2,3}}
  56: {{3},{1,3},{2,3}}

Intersecting set systems are A051185 (not-covering) or A305843 (covering).
BII-numbers of set-systems with empty intersection are A326911.
Cf. A006058, A048793, A326031, A326875, A326912, A326913.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[0,100],stableQ[bpe/@bpe[#],Intersection[#1,#2]=={}&]&]

A337694 Numbers with no two relatively prime prime indices.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 129, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167, 169, 171, 173, 179, 181, 183, 185, 189, 191, 193, 197, 199

Offset: 1

Gus Wiseman, Sep 23 2020

nonn

First differs from A305078 in having 1 and lacking 195.
First differs from A305103 in having 1 and 169 and lacking 195.
First differs from A328336 in lacking 897, with prime indices (2,6,9).
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.
Also Heinz numbers of integer partitions in which no two parts are relatively prime. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
   1: {}      37: {12}     79: {22}      121: {5,5}
   3: {2}     39: {2,6}    81: {2,2,2,2} 125: {3,3,3}
   5: {3}     41: {13}     83: {23}      127: {31}
   7: {4}     43: {14}     87: {2,10}    129: {2,14}
   9: {2,2}   47: {15}     89: {24}      131: {32}
  11: {5}     49: {4,4}    91: {4,6}     133: {4,8}
  13: {6}     53: {16}     97: {25}      137: {33}
  17: {7}     57: {2,8}   101: {26}      139: {34}
  19: {8}     59: {17}    103: {27}      147: {2,4,4}
  21: {2,4}   61: {18}    107: {28}      149: {35}
  23: {9}     63: {2,2,4} 109: {29}      151: {36}
  25: {3,3}   65: {3,6}   111: {2,12}    157: {37}
  27: {2,2,2} 67: {19}    113: {30}      159: {2,16}
  29: {10}    71: {20}    115: {3,9}     163: {38}
  31: {11}    73: {21}    117: {2,2,6}   167: {39}

Robert Israel, Table of n, a(n) for n = 1..10000

A200976 and A328673 count these partitions.
A302696 and A302569 are pairwise coprime instead of pairwise non-coprime.
A318719 is the squarefree case.
A328867 looks at distinct prime indices.
A337666 is the version for standard compositions.
A101268 counts pairwise coprime or singleton compositions.
A318717 counts strict pairwise non-coprime partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A335236 ranks compositions neither a singleton nor pairwise coprime.
A337462 counts pairwise coprime compositions.
A337667 counts pairwise non-coprime compositions.
Cf. A051185, A051424, A056239, A112798, A220377, A284825, A302797, A303282, A305843, A319752, A336737, A337599, A337604, A337605.

filter:= proc(n) local F,i,j,np;
  if n::even and n>2 then return false fi;
  F:= map(t -> numtheory:-pi(t[1]), ifactors(n)[2]);
  np:= nops(F);
  for i from 1 to np-1 do
    for j from i+1 to np do
      if igcd(F[i],F[j])=1 then return false fi
  od od;
  true
end proc:
select(filter, [$1..300]); # Robert Israel, Oct 06 2020

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stabQ[u_,Q_]:=Array[#1==#2||!Q[u[[#1]],u[[#2]]]&,{Length[u],Length[u]},1,And];
Select[Range[100],stabQ[primeMS[#],CoprimeQ]&]

A319774 Number of intersecting set systems spanning n vertices whose dual is also an intersecting set system.

Original entry on oeis.org

1, 1, 2, 14, 814, 1174774, 909125058112, 291200434263385001951232

Offset: 0

Gus Wiseman, Sep 27 2018

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.

A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.

			The a(3) = 14 set systems:
   {{1},{1,2},{1,2,3}}
   {{1},{1,3},{1,2,3}}
   {{2},{1,2},{1,2,3}}
   {{2},{2,3},{1,2,3}}
   {{3},{1,3},{1,2,3}}
   {{3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3}}
   {{1,2},{1,3},{1,2,3}}
   {{1,2},{2,3},{1,2,3}}
   {{1,3},{2,3},{1,2,3}}
   {{1},{1,2},{1,3},{1,2,3}}
   {{2},{1,2},{2,3},{1,2,3}}
   {{3},{1,3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3},{1,2,3}}

Cf. A007716, A281116, A283877, A305854, A306006,  A316980, A316983, A317757, A319616.
Cf. A319752, A319765, A319766, A319767, A319768, A319769.
Intersecting set-systems are A051185.
The unlabeled multiset partition version is A319773.
The covering case is A327037.
The version without strict dual is A327038.
Cointersecting set-systems are A327039.
The BII-numbers of these set-systems are A327061.
Cf. A003465, A305843, A305844, A326854, A327020, A327040, A327052.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&UnsameQ@@dual[#]&&stableQ[#,Intersection[#1,#2]=={}&]&&stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,3}] (* Gus Wiseman, Aug 19 2019 *)

a(6)-a(7) from Christian Sievers, Aug 18 2024

A337666 Numbers k such that any two parts of the k-th composition in standard order (A066099) have a common divisor > 1.

Original entry on oeis.org

0, 2, 4, 8, 10, 16, 32, 34, 36, 40, 42, 64, 128, 130, 136, 138, 160, 162, 168, 170, 256, 260, 288, 292, 512, 514, 520, 522, 528, 544, 546, 552, 554, 640, 642, 648, 650, 672, 674, 680, 682, 1024, 2048, 2050, 2052, 2056, 2058, 2080, 2082, 2084, 2088, 2090, 2176

Offset: 1

Gus Wiseman, Oct 05 2020

nonn

Differs from A291165 in having 1090535424, corresponding to the composition (6,10,15).
This is a ranking sequence for pairwise non-coprime compositions.
The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence together with the corresponding compositions begins:
       0: ()          138: (4,2,2)       546: (4,4,2)
       2: (2)         160: (2,6)         552: (4,2,4)
       4: (3)         162: (2,4,2)       554: (4,2,2,2)
       8: (4)         168: (2,2,4)       640: (2,8)
      10: (2,2)       170: (2,2,2,2)     642: (2,6,2)
      16: (5)         256: (9)           648: (2,4,4)
      32: (6)         260: (6,3)         650: (2,4,2,2)
      34: (4,2)       288: (3,6)         672: (2,2,6)
      36: (3,3)       292: (3,3,3)       674: (2,2,4,2)
      40: (2,4)       512: (10)          680: (2,2,2,4)
      42: (2,2,2)     514: (8,2)         682: (2,2,2,2,2)
      64: (7)         520: (6,4)        1024: (11)
     128: (8)         522: (6,2,2)      2048: (12)
     130: (6,2)       528: (5,5)        2050: (10,2)
     136: (4,4)       544: (4,6)        2052: (9,3)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

A337604 counts these compositions of length 3.
A337667 counts these compositions.
A337694 is the version for Heinz numbers of partitions.
A337696 is the strict case.
A051185 and A305843 (covering) count pairwise intersecting set-systems.
A101268 counts pairwise coprime or singleton compositions.
A200976 and A328673 count pairwise non-coprime partitions.
A318717 counts strict pairwise non-coprime partitions.
A327516 counts pairwise coprime partitions.
A335236 ranks compositions neither a singleton nor pairwise coprime.
A337462 counts pairwise coprime compositions.
All of the following pertain to compositions in standard order (A066099):
- A000120 is length.
- A070939 is sum.
- A124767 counts runs.
- A233564 ranks strict compositions.
- A272919 ranks constant compositions.
- A291166 appears to rank relatively prime compositions.
- A326674 is greatest common divisor.
- A333219 is Heinz number.
- A333227 ranks coprime (Mathematica definition) compositions.
- A333228 ranks compositions with distinct parts coprime.
- A335235 ranks singleton or coprime compositions.
Cf. A082024, A284825, A305713, A319752, A319786, A327039, A327040, A336737, A337599, A337605.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
Select[Range[0,1000],stabQ[stc[#],CoprimeQ]&]

A327020 Number of antichains covering n vertices where every two vertices appear together in some edge (cointersecting).

Original entry on oeis.org

1, 1, 1, 2, 17, 1451, 3741198

Offset: 0

Gus Wiseman, Aug 17 2019

A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges, The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. An antichain is a set of sets, none of which is a subset of any other. This sequence counts antichains with union {1..n} whose dual is pairwise intersecting.

			The a(0) = 1 through a(4) = 17 antichains:
  {}  {{1}}  {{12}}  {{123}}         {{1234}}
                     {{12}{13}{23}}  {{12}{134}{234}}
                                     {{13}{124}{234}}
                                     {{14}{123}{234}}
                                     {{23}{124}{134}}
                                     {{24}{123}{134}}
                                     {{34}{123}{124}}
                                     {{123}{124}{134}}
                                     {{123}{124}{234}}
                                     {{123}{134}{234}}
                                     {{124}{134}{234}}
                                     {{12}{13}{14}{234}}
                                     {{12}{23}{24}{134}}
                                     {{13}{23}{34}{124}}
                                     {{14}{24}{34}{123}}
                                     {{123}{124}{134}{234}}
                                     {{12}{13}{14}{23}{24}{34}}

Covering, intersecting antichains are A305844.
Covering, T1 antichains are A319639.
Cointersecting set-systems are A327039.
Covering, cointersecting set-systems are A327040.
Covering, cointersecting set-systems are A327051.
The non-covering version is A327057.
Covering, intersecting, T1 set-systems are A327058.
Unlabeled cointersecting antichains of multisets are A327060.
Cf. A003465, A305843, A319765, A326853, A327037, A327038, A327053.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
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]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],Union@@#==Range[n]&&stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,4}]

Inverse binomial transform of A327057.

A326912 BII-numbers of pairwise intersecting set-systems with empty intersection.

Original entry on oeis.org

0, 52, 116, 772, 832, 836, 1072, 1076, 1136, 1140, 1796, 1856, 1860, 2320, 2368, 2384, 2592, 2624, 2656, 2880, 3088, 3104, 3120, 3136, 3152, 3168, 3184, 3344, 3392, 3408, 3616, 3648, 3680, 3904, 4132, 4148, 4196, 4212, 4612, 4640, 4644, 4672, 4676, 4704, 4708

Offset: 1

Gus Wiseman, Aug 04 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

			The sequence of all pairwise intersecting set-systems with empty intersection, together with their BII-numbers, begins:
     0: {}
    52: {{1,2},{1,3},{2,3}}
   116: {{1,2},{1,3},{2,3},{1,2,3}}
   772: {{1,2},{1,4},{2,4}}
   832: {{1,2,3},{1,4},{2,4}}
   836: {{1,2},{1,2,3},{1,4},{2,4}}
  1072: {{1,3},{2,3},{1,2,4}}
  1076: {{1,2},{1,3},{2,3},{1,2,4}}
  1136: {{1,3},{2,3},{1,2,3},{1,2,4}}
  1140: {{1,2},{1,3},{2,3},{1,2,3},{1,2,4}}
  1796: {{1,2},{1,4},{2,4},{1,2,4}}
  1856: {{1,2,3},{1,4},{2,4},{1,2,4}}
  1860: {{1,2},{1,2,3},{1,4},{2,4},{1,2,4}}
  2320: {{1,3},{1,4},{3,4}}
  2368: {{1,2,3},{1,4},{3,4}}
  2384: {{1,3},{1,2,3},{1,4},{3,4}}
  2592: {{2,3},{2,4},{3,4}}
  2624: {{1,2,3},{2,4},{3,4}}
  2656: {{2,3},{1,2,3},{2,4},{3,4}}
  2880: {{1,2,3},{1,4},{2,4},{3,4}}

Cf. A048793, A051185, A305843, A317752, A317755, A317757, A319077, A326031, A326910, A326911.

Cf. A318715, A319759, A319762, A319763, A319764.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,1000],(#==0||Intersection@@bpe/@bpe[#]=={})&&stableQ[bpe/@bpe[#],Intersection[#1,#2]=={}&]&]

A202425 Number of partitions of n into parts having pairwise common factors but no overall common factor.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 3, 0, 0, 1, 6, 0, 5, 0, 2, 2, 9, 0, 8, 2, 4, 3, 16, 0, 22, 5, 6, 5, 19, 2, 35, 8, 14, 6, 44, 4, 55, 13, 16, 19, 64, 6, 82, 17, 39, 31, 108, 10, 105, 40, 66, 46, 161, 14, 182, 61, 97, 72, 207, 37, 287, 85, 144, 93, 357, 59

Offset: 31

Alois P. Heinz, Dec 19 2011

nonn

			a(31) = 1: [6,10,15] = [2*3,2*5,3*5].
a(37) = 2: [6,6,10,15], [10,12,15].
a(41) = 3: [6,10,10,15], [6,15,20], [6,14,21].
a(47) = 6: [6,6,10,10,15], [10,10,12,15], [6,6,15,20], [12,15,20], [6,6,14,21], [12,14,21].
a(49) = 5: [6,6,6,6,10,15], [6,6,10,12,15], [10,12,12,15], [6,10,15,18], [10,15,24].

Fausto A. C. Cariboni, Table of n, a(n) for n = 31..400 (terms 31..251 from Alois P. Heinz)

Cf. A202385, A018783.
The version with only distinct parts compared is A328672.
The Heinz numbers of these partitions are A328868.
The strict case is A202385, which is essentially the same as A318715.
The version for non-isomorphic multiset partitions is A319759.
The version for set-systems is A326364.
Intersecting partitions are A200976.
Cf. A000837, A305148, A305843, A306006, A318717, A328673, A328867.

with(numtheory):
w:= (m, h)-> mul(`if`(j>=h, 1, j), j=factorset(m)):
b:= proc(n, i, g, s) option remember; local j, ok, si;
      if n<0 then 0
    elif n=0 then `if`(g>1, 0, 1)
    elif i<2 or member(1, s) then 0
    else ok:= evalb(i<=n);
         si:= map(x->w(x, i), s);
         for j in s while ok do ok:= igcd(i, j)>1 od;
         b(n, i-1, g, si) +`if`(ok, add(b(n-t*i, i-1, igcd(i, g),
                      si union {w(i,i)} ), t=1..iquo(n, i)), 0)
      fi
    end:
a:= n-> b(n, n, 0, {}):
seq(a(n), n=31..100);

w[m_, h_] := Product[If[j >= h, 1, j], {j, FactorInteger[m][[All, 1]]}]; b[n_, i_, g_, s_] := b[n, i, g, s] = Module[{j, ok, si}, Which[n<0, 0, n == 0, If[g>1, 0, 1], i<2 || MemberQ[s, 1], 0, True, ok = (i <= n); si = w[#, i]& /@ s; Do[If[ok, ok = (GCD[i, j]>1)], {j, s}]; b[n, i-1, g, si] + If[ok, Sum[b[n-t*i, i-1, GCD[i, g], si ~Union~ {w[i, i]}], {t, 1, Quotient[n, i]}], 0]]]; a[n_] := b[n, n, 0, {}]; Table[a[n], {n, 31, 100}] (* Jean-François Alcover, Feb 16 2017, translated from Maple *)
Table[Length[Select[IntegerPartitions[n],GCD@@#==1&&And@@(GCD[##]>1&)@@@Tuples[#,2]&]],{n,0,40}] (* Gus Wiseman, Nov 04 2019 *)

a(n > 0) = A328672(n) - 1. - Gus Wiseman, Nov 04 2019

A318720 Numbers k such that there exists a strict relatively prime factorization of k in which no pair of factors is relatively prime.

Original entry on oeis.org

900, 1764, 1800, 2700, 3528, 3600, 4356, 4500, 4900, 5292, 5400, 6084, 6300, 7056, 7200, 8100, 8712, 8820, 9000, 9800, 9900, 10404, 10584, 10800, 11025, 11700, 12100, 12168, 12348, 12600, 12996, 13068, 13500, 14112, 14400, 14700, 15300, 15876, 16200, 16900

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

From Amiram Eldar, Nov 01 2020: (Start)
Also, numbers with more than two non-unitary prime divisors, i.e., numbers k such that A056170(k) > 2, or equivalently, numbers divisible by the squares of three distinct primes.
The complement of the union of A005117, A190641 and A338539.
The asymptotic density of this sequence is 1 - 6/Pi^2 - (6/Pi^2)*A154945 - (3/Pi^2)*(A154945^2 - A324833) = 0.0033907041... (End)

			900 is in the sequence because the factorization 900 = (6*10*15) is relatively prime (since the GCD of (6,10,15) is 1) but each of the pairs (6,10), (6,15), (10,15) has a common divisor > 1. Larger examples are:
1800 = (6*15*20) = (10*12*15).
9900 = (6*10*165) = (6*15*110) = (10*15*66).
5400 = (6*20*45) = (10*12*45) = (10*15*36) = (15*18*20).
60 is not in the sequence because all its possible factorizations (4 * 15, 3 * 4 * 5, etc.) contain at least one pair that is coprime, if not more than one prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001055, A001221, A001222, A007716, A045778, A051185, A078374, A281116, A303140, A303283, A305843, A305854, A317748, A318715, A318717, A318721.

Cf. A005117, A036785, A056170, A154945, A190641, A324833, A338539.

strfacs[n_] := If[n <= 1, {{}}, Join@@Table[(Prepend[#1, d] &)/@Select[strfacs[n/d], Min@@#1 > d &], {d, Rest[Divisors[n]]}]]; Select[Range[10000], Function[n, Select[strfacs[n], And[GCD@@# == 1, And@@(GCD[##] > 1 &)@@@Select[Tuples[#, 2], Less@@# &]] &] != {}]]
Select[Range[20000], Count[FactorInteger[#][[;;,2]], ?(#1 > 1 &)] > 2 &] (* _Amiram Eldar, Nov 01 2020 *)

A327037 Number of pairwise intersecting set-systems covering n vertices where every two vertices appear together in some edge (cointersecting).

Original entry on oeis.org

1, 1, 3, 21, 913, 1183295, 909142733955, 291200434282476769116160

Offset: 0

Gus Wiseman, Aug 17 2019

A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. This sequence counts pairwise intersecting, covering set-systems that are cointersecting, meaning their dual is pairwise intersecting.

			The a(0) = 1 through a(3) = 21 set-systems:
  {}  {{1}}  {{1,2}}      {{1,2,3}}
             {{1},{1,2}}  {{1},{1,2,3}}
             {{2},{1,2}}  {{2},{1,2,3}}
                          {{3},{1,2,3}}
                          {{1,2},{1,2,3}}
                          {{1,3},{1,2,3}}
                          {{2,3},{1,2,3}}
                          {{1},{1,2},{1,2,3}}
                          {{1},{1,3},{1,2,3}}
                          {{1,2},{1,3},{2,3}}
                          {{2},{1,2},{1,2,3}}
                          {{2},{2,3},{1,2,3}}
                          {{3},{1,3},{1,2,3}}
                          {{3},{2,3},{1,2,3}}
                          {{1,2},{1,3},{1,2,3}}
                          {{1,2},{2,3},{1,2,3}}
                          {{1,3},{2,3},{1,2,3}}
                          {{1},{1,2},{1,3},{1,2,3}}
                          {{2},{1,2},{2,3},{1,2,3}}
                          {{3},{1,3},{2,3},{1,2,3}}
                          {{1,2},{1,3},{2,3},{1,2,3}}

Intersecting covering set-systems are A305843.
The unlabeled multiset partition version is A319765.
The case where the dual is strict is A319774.
The BII-numbers of these set-systems are A326912.
The non-covering version is A327038.
Cointersectng covering set-systems are A327040.
Cf. A003465, A319767, A326853, A326854, A326910.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
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]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Intersection[#1,#2]=={}&],Union@@#==Range[n]&&stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,4}]

Inverse binomial transform of A327038.

a(6)-a(7) from Christian Sievers, Aug 18 2024

cp's OEIS Frontend

A337667 Number of compositions of n where any two parts have a common divisor > 1.

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A326910 BII-numbers of pairwise intersecting set-systems.

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A337694 Numbers with no two relatively prime prime indices.

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A319774 Number of intersecting set systems spanning n vertices whose dual is also an intersecting set system.

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A337666 Numbers k such that any two parts of the k-th composition in standard order (A066099) have a common divisor > 1.

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A327020 Number of antichains covering n vertices where every two vertices appear together in some edge (cointersecting).

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A326912 BII-numbers of pairwise intersecting set-systems with empty intersection.

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A202425 Number of partitions of n into parts having pairwise common factors but no overall common factor.

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