A051185 -id:A051185 - cp's OEIS frontend

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

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 *)

A327038 Number of pairwise intersecting set-systems covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).

Original entry on oeis.org

1, 2, 6, 34, 1020, 1188106, 909149847892, 291200434288840793135801

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 set-systems that are cointersecting, meaning their dual is pairwise intersecting.

			The a(0) = 1 through a(2) = 6 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1,2}}
             {{1},{1,2}}
             {{2},{1,2}}
The a(3) = 34 set-systems:
  {}  {{1}}    {{1}{12}}    {{1}{12}{123}}   {{1}{12}{13}{123}}
      {{2}}    {{1}{13}}    {{1}{13}{123}}   {{2}{12}{23}{123}}
      {{3}}    {{2}{12}}    {{12}{13}{23}}   {{3}{13}{23}{123}}
      {{12}}   {{2}{23}}    {{2}{12}{123}}   {{12}{13}{23}{123}}
      {{13}}   {{3}{13}}    {{2}{23}{123}}
      {{23}}   {{3}{23}}    {{3}{13}{123}}
      {{123}}  {{1}{123}}   {{3}{23}{123}}
               {{2}{123}}   {{12}{13}{123}}
               {{3}{123}}   {{12}{23}{123}}
               {{12}{123}}  {{13}{23}{123}}
               {{13}{123}}
               {{23}{123}}

Intersecting set-systems are A051185.
The unlabeled multiset partition version is A319765.
The BII-numbers of these set-systems are A326912.
The covering case is A327037.
Cointersecting set-systems are A327039.
The case where the dual is strict is A327040.
Cf. A058891, A319767, A319774, A326854, A327052.

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]=={}&],stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,4}]

Binomial transform of A327037.

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

A327052 Number of T_0 (costrict) set-systems covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).

Original entry on oeis.org

1, 2, 6, 75, 24981, 2077072342, 9221293211115589902, 170141182628636920748880864929055912851

Offset: 0

Gus Wiseman, Aug 18 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 set-systems whose dual is strict and pairwise intersecting.

			The a(0) = 1 through a(2) = 6 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1},{1,2}}
             {{2},{1,2}}
             {{1},{2},{1,2}}

The unlabeled multiset partition version is A319760.
The non-T_0 version is A327039.
The covering case is A327053.
Cf. A051185, A319752, A319774, A327038, A327040.

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}]],UnsameQ@@dual[#]&&stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,3}]

Binomial transform of A327053.

a(5)-a(7) from Christian Sievers, Feb 04 2024

A051184 Number of 7-element intersecting families of an n-element set.

Original entry on oeis.org

0, 0, 0, 0, 80, 169125, 71102400, 18047221707, 3623784887164, 638772147728325, 103751227132038920, 15931275037246743999, 2348130220089143792148, 335520750110815538499945, 46803828588394634589433120

Offset: 0

Vladeta Jovovic, Goran Kilibarda

nonn

V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

Cf. A036239, A051180-A051185.

1/7! (128^n - 21*96^n + 105*80^n - 35*72^n + 105*68^n - 42*66^n + 7*65^n - 476*64^n - 630*60^n + 1785*56^n + 315*54^n - 210*52^n - 105*51^n + 1260*50^n - 105*49^n - 1575*48^n - 2520*46^n - 105*45^n + 1638*44^n + 840*43^n - 6615*42^n + 1050*41^n + 4130*40^n - 1890*39^n + 14595*38^n + 2835*37^n - 7945*36^n - 1554*35^n - 18711*34^n - 12572*33^n + 24710*32^n + 4620*31^n + 560*30^n + 25995*29^n - 16905*28^n - 13545*27^n - 6510*26^n - 42945*25^n + 12005*24^n + 102011*23^n - 4648*22^n - 87780*21^n - 15785*20^n + 43120*19^n + 21364*18^n + 4200*17^n - 37205*16^n - 17105*15^n + 36386*14^n + 28644*13^n - 57603*12^n + 24150*11^n + 4585*10^n - 16289*9^n + 20943*8^n - 12754*7^n - 287*6^n + 4137*5^n - 3388*4^n + 1764*3^n + 720*2^n - 720)

A305999 Number of unlabeled spanning intersecting set-systems on n vertices with no singletons.

Original entry on oeis.org

1, 0, 1, 6, 76, 12916

Offset: 0

Gus Wiseman, Jun 16 2018

An intersecting set-system S is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection. S is spanning if every vertex is contained in some edge. A singleton is an edge containing only one vertex.

			Non-isomorphic representative of the a(3) = 6 set-systems:
{{1,2,3}}
{{1,3},{2,3}}
{{2,3},{1,2,3}}
{{1,2},{1,3},{2,3}}
{{1,3},{2,3},{1,2,3}}
{{1,2},{1,3},{2,3},{1,2,3}}

Cf. A001206, A051185, A048143, A261006, A058891, A261005, A304998, A305854-A305857, A305935, A306000, A306001, A306008.

a(n) = A306001(n) - A306001(n-1) for n > 0. - Andrew Howroyd, Aug 12 2019

a(5) from Andrew Howroyd, Aug 12 2019

A051182 Number of 5-element intersecting families of an n-element set.

Original entry on oeis.org

0, 0, 0, 0, 371, 38163, 2236504, 103998636, 4289058501, 164693276181, 6034793020298, 213993130915542, 7407880110115111, 251837583669470799, 8443568934653875932, 280082506996725346368

Offset: 0

Vladeta Jovovic, Goran Kilibarda

nonn

V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

Cf. A036239, A051180-A051185.

1/5!(32^n-10*24^n+30*20^n-5*18^n+5*17^n-80*16^n-30*15^n+135*14^n+30*13^n-80*12^n-2*11^n+10*10^n-100*9^n+240*8^n-160*7^n-44*6^n+95*5^n-85*4^n+50*3^n+24*2^n-24).

A051183 Number of 6-element intersecting families of an n-element set.

Original entry on oeis.org

0, 0, 0, 0, 230, 91993, 14037879, 1509286261, 136653987232, 11209147489701, 862949794999193, 63573922606869037, 4535012297248660194, 315713834759742768349, 21570075957885603579067

Offset: 0

Vladeta Jovovic, Goran Kilibarda

nonn

V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

Cf. A036239, A051180-A051185.

1/6! (64^n - 15*48^n + 60*40^n - 15*36^n + 30*34^n - 6*33^n - 215*32^n - 180*30^n + 585*28^n + 45*27^n + 60*26^n + 150*25^n - 510*24^n - 360*23^n + 168*22^n - 585*21^n + 795*20^n + 1665*19^n - 1890*18^n - 2175*17^n + 3305*16^n + 1775*15^n - 3795*14^n - 870*13^n + 3123*12^n - 1075*11^n - 495*10^n + 1460*9^n - 2245*8^n + 1424*7^n + 150*6^n - 590*5^n + 499*4^n - 274*3^n - 120*2^n + 120)

A318749 Number of pairwise relatively nonprime strict factorizations of n (no two factors are coprime).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 5, 2, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 2, 1, 1, 1, 7, 1, 2, 2, 3, 1, 1, 1, 3, 1

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Oct 08 2018

			The a(96) = 7 factorizations are (96), (2*48), (4*24), (6*16), (8*12), (2*4*12), (2*6*8).
The a(480) = 18 factorizations:
  (480)
  (2*240) (4*120) (6*80) (8*60) (10*48) (12*40) (16*30) (20*24)
  (2*4*60) (2*6*40) (2*8*30) (2*10*24) (2*12*20) (4*6*20) (4*10*12) (6*8*10)
  (2*4*6*10)

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A001055, A045778, A051185, A281116, A303282, A305843, A305854, A317748, A318715, A318717, A318721.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[strfacs[n/d],Min@@#1>d&],{d,Rest[Divisors[n]]}]];
Table[Length[Select[strfacs[n],And@@(GCD[##]>1&)@@@Select[Tuples[#,2],Less@@#&]&]],{n,50}]

A318749(n, m=n, facs=List([])) = if(1==n, (1!=gcd(Vec(facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A318749(n/d, d-1, newfacs))); (s)); \\ Antti Karttunen, Oct 08 2018

More terms from Antti Karttunen, Oct 08 2018

A326365 Number of intersecting antichains with empty intersection (meaning there is no vertex in common to all the edges) covering n vertices.

Original entry on oeis.org

1, 0, 0, 1, 23, 1834, 1367903, 229745722873, 423295077919493525420

Offset: 0

Gus Wiseman, Jul 01 2019

Covering means there are no isolated vertices. A set system (set of sets) is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint.

			The a(4) = 23 intersecting antichains with empty intersection:
  {{1,2},{1,3},{2,3,4}}
  {{1,2},{1,4},{2,3,4}}
  {{1,2},{2,3},{1,3,4}}
  {{1,2},{2,4},{1,3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,3},{2,3},{1,2,4}}
  {{1,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{1,2,3}}
  {{1,4},{3,4},{1,2,3}}
  {{2,3},{2,4},{1,3,4}}
  {{2,3},{3,4},{1,2,4}}
  {{2,4},{3,4},{1,2,3}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,2,4},{2,3,4}}
  {{1,4},{1,2,3},{2,3,4}}
  {{2,3},{1,2,4},{1,3,4}}
  {{2,4},{1,2,3},{1,3,4}}
  {{3,4},{1,2,3},{1,2,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{2,3},{2,4},{1,3,4}}
  {{1,3},{2,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{3,4},{1,2,3}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

Intersecting antichain covers are A305844.
Intersecting covers with empty intersection are A326364.
Antichain covers with empty intersection are A305001.
The binomial transform is the non-covering case A326366.
Covering, intersecting antichains with empty intersection are A326365.
Cf. A006126, A007363, A014466, A051185, A058891, A305843, A307249, A318128, A318129, A326361, A326362, A326363.

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]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&],And[Union@@#==Range[n],#=={}||Intersection@@#=={}]&]],{n,0,4}]

a(7)-a(8) from Andrew Howroyd, Aug 14 2019

A326366 Number of intersecting antichains of nonempty subsets of {1..n} with empty intersection (meaning there is no vertex in common to all the edges).

Original entry on oeis.org

1, 1, 1, 2, 28, 1960, 1379273, 229755337549, 423295079757497714059

Offset: 0

Gus Wiseman, Jul 01 2019

A set system (set of sets) is an antichain if no edge is a subset of any other, and is intersecting if no two edges are disjoint.

			The a(0) = 1 through a(4) = 28 intersecting antichains with empty intersection:
  {}  {}  {}  {}              {}
              {{12}{13}{23}}  {{12}{13}{23}}
                              {{12}{14}{24}}
                              {{13}{14}{34}}
                              {{23}{24}{34}}
                              {{12}{13}{234}}
                              {{12}{14}{234}}
                              {{12}{23}{134}}
                              {{12}{24}{134}}
                              {{13}{14}{234}}
                              {{13}{23}{124}}
                              {{13}{34}{124}}
                              {{14}{24}{123}}
                              {{14}{34}{123}}
                              {{23}{24}{134}}
                              {{23}{34}{124}}
                              {{24}{34}{123}}
                              {{12}{134}{234}}
                              {{13}{124}{234}}
                              {{14}{123}{234}}
                              {{23}{124}{134}}
                              {{24}{123}{134}}
                              {{34}{123}{124}}
                              {{12}{13}{14}{234}}
                              {{12}{23}{24}{134}}
                              {{13}{23}{34}{124}}
                              {{14}{24}{34}{123}}
                              {{123}{124}{134}{234}}

The case with empty edges allowed is A326375.
Intersecting antichains of nonempty sets are A001206.
Intersecting set systems with empty intersection are A326373.
Antichains of nonempty sets with empty intersection are A006126 or A307249.
The inverse binomial transform is the covering case A326365.
Cf. A007363, A014466, A051185, A058891, A305001, A305843, A305844, A318128, A318129, A326361, A326362, A326363, A326364.

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]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&],#=={}||Intersection@@#=={}&]],{n,0,4}]

a(n) = A326375(n) - 1.

a(n) = A001206(n+1) + A307249(n) - A014466(n). - Andrew Howroyd, Aug 14 2019

a(7)-a(8) from Andrew Howroyd, Aug 14 2019

cp's OEIS Frontend

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

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A327038 Number of pairwise intersecting set-systems covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).

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A327052 Number of T_0 (costrict) set-systems covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).

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A051184 Number of 7-element intersecting families of an n-element set.

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A305999 Number of unlabeled spanning intersecting set-systems on n vertices with no singletons.

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A051182 Number of 5-element intersecting families of an n-element set.

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A051183 Number of 6-element intersecting families of an n-element set.

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A318749 Number of pairwise relatively nonprime strict factorizations of n (no two factors are coprime).

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A326365 Number of intersecting antichains with empty intersection (meaning there is no vertex in common to all the edges) covering n vertices.

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