A318715 -id:A318715 - cp's OEIS frontend

A319760 Number of non-isomorphic intersecting strict multiset partitions (sets of multisets) of weight n.

1, 1, 2, 5, 11, 26, 68, 162, 423, 1095, 2936

Offset: 0

Gus Wiseman, Sep 27 2018

A multiset partition is intersecting if no two parts are disjoint. 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(1) = 1 through a(4) = 11 strict multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1,2}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,2,3}}
   {{1},{1,1}}
   {{2},{1,2}}
4: {{1,1,1,1}}
   {{1,1,2,2}}
   {{1,2,2,2}}
   {{1,2,3,3}}
   {{1,2,3,4}}
   {{1},{1,1,1}}
   {{1},{1,2,2}}
   {{2},{1,2,2}}
   {{3},{1,2,3}}
   {{1,2},{2,2}}
   {{1,3},{2,3}}

Cf. A007716, A283877, A305854, A306006, A316980, A318715, A318717.

Cf. A319752, A319755, A319759, A319765, A319779, A319787, A319789.

A328672 Number of integer partitions of n with relatively prime parts in which no two distinct parts are relatively prime.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 4, 1, 1, 2, 7, 1, 6, 1, 3, 3, 10, 1, 9, 3, 5, 4, 17, 1, 23, 6, 7, 6, 20, 3, 36, 9, 15, 7, 45, 5, 56, 14, 17, 20, 65, 7, 83, 18, 40

Offset: 0

Gus Wiseman, Oct 29 2019

nonn

Positions of terms greater than 1 are {31, 37, 41, 43, 46, 47, 49, ...}.

A partition with no two distinct parts relatively prime is said to be intersecting.

			Examples:
  a(31) = 2:         a(46) = 2:
    (15,10,6)          (15,15,10,6)
    (1^31)             (1^46)
  a(37) = 3:         a(47) = 7:
    (15,12,10)         (20,15,12)
    (15,10,6,6)        (21,14,12)
    (1^37)             (20,15,6,6)
  a(41) = 4:           (21,14,6,6)
    (20,15,6)          (15,12,10,10)
    (21,14,6)          (15,10,10,6,6)
    (15,10,10,6)       (1^47)
    (1^41)           a(49) = 6:
  a(43) = 4:           (24,15,10)
    (18,15,10)         (18,15,10,6)
    (15,12,10,6)       (15,12,12,10)
    (15,10,6,6,6)      (15,12,10,6,6)
    (1^43)             (15,10,6,6,6,6)
                       (1^39)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..400

The Heinz numbers of these partitions are A328679.
The strict case is A318715.
The version for non-isomorphic multiset partitions is A319759.
Relatively prime partitions are A000837.
Intersecting partitions are A328673.
Cf. A078374, A285573, A289509, A291166, A303140, A305148, A316476, A326910, A326912.

Table[Length[Select[IntegerPartitions[n],GCD@@#==1&&And[And@@(GCD[##]>1&)@@@Subsets[Union[#],{2}]]&]],{n,0,32}]

a(n > 0) = A202425(n) + 1.

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

A328868 Heinz numbers of integer partitions with no two (not necessarily distinct) parts relatively prime, but with no divisor in common to all of the parts.

Original entry on oeis.org

17719, 40807, 43381, 50431, 74269, 83143, 101543, 105703, 116143, 121307, 123469, 139919, 140699, 142883, 171613, 181831, 185803, 191479, 203557, 205813, 211381, 213239, 215267, 219271, 230347, 246703, 249587, 249899, 279371, 286897, 289007, 296993, 300847

Offset: 1

Gus Wiseman, Oct 30 2019

nonn

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:
   17719: {6,10,15}
   40807: {6,14,21}
   43381: {6,15,20}
   50431: {10,12,15}
   74269: {6,10,45}
   83143: {10,15,18}
  101543: {6,21,28}
  105703: {6,15,40}
  116143: {12,14,21}
  121307: {10,15,24}
  123469: {12,15,20}
  139919: {6,15,50}
  140699: {6,22,33}
  142883: {6,10,75}
  171613: {6,14,63}
  181831: {6,20,45}
  185803: {10,14,35}
  191479: {14,18,21}
  203557: {15,18,20}
  205813: {10,15,36}
  211381: {10,12,45}
  213239: {6,15,70}
  215267: {6,10,105}
  219271: {6,26,39}
  230347: {6,6,10,15}

These are the Heinz numbers of the partitions counted by A202425.
Terms of A328679 that are not powers of 2.
The strict case is A318716 (preceded by 2).
A ranking using binary indices (instead of prime indices) is A326912.
Heinz numbers of relatively prime partitions are A289509.
Cf. A000837, A056239, A112798, A200976, A291166, A302796, A316476, A318715, A319752, A319759, A328336, A328672, A328677, A328867.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dv=Select[Range[100000],GCD@@primeMS[#]==1&&And[And@@(GCD[##]>1&)@@@Tuples[Union[primeMS[#]],2]]&]

A319762 Number of non-isomorphic intersecting set multipartitions (multisets of sets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 4, 9, 24

Offset: 0

Gus Wiseman, Sep 27 2018

A set multipartition is intersecting if no two parts are disjoint. The weight of a set multipartition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(6) = 1 through a(9) = 9 set multipartitions:
6: {{1,2},{1,3},{2,3}}
7: {{1,3},{1,4},{2,3,4}}
8: {{1,2},{1,3,4},{2,3,4}}
   {{1,4},{1,5},{2,3,4,5}}
   {{2,4},{1,2,5},{3,4,5}}
   {{1,2},{1,3},{2,3},{2,3}}
9: {{1,3},{1,4,5},{2,3,4,5}}
   {{1,5},{1,6},{2,3,4,5,6}}
   {{2,5},{1,2,6},{3,4,5,6}}
   {{1,2,3},{2,4,5},{3,4,5}}
   {{1,3,5},{2,3,6},{4,5,6}}
   {{1,2},{1,3},{1,4},{2,3,4}}
   {{1,2},{1,3},{2,3},{1,2,3}}
   {{1,3},{1,4},{1,4},{2,3,4}}
   {{1,3},{1,4},{3,4},{2,3,4}}

Cf. A007716, A049311, A281116, A283877, A305854, A306006, A316980, A317757, A318715, A318717.

Cf. A319752, A319759, A319763, A319764, A319765, A319779, A319781.

A319763 Number of non-isomorphic strict intersecting multiset partitions (sets of multisets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 2, 12, 46, 181

Offset: 0

Gus Wiseman, Sep 27 2018

A multiset partition is intersecting if no two parts are disjoint. 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(6) = 1 through a(8) = 12 multiset partitions:
6: {{1,2},{1,3},{2,3}}
7: {{1,2},{1,3},{2,3,3}}
   {{1,3},{1,4},{2,3,4}}
8: {{1,2},{1,3},{2,2,3,3}}
   {{1,2},{1,3},{2,3,3,3}}
   {{1,2},{1,3},{2,3,4,4}}
   {{1,2},{1,3,3},{2,3,3}}
   {{1,2},{1,3,4},{2,3,4}}
   {{1,3},{1,4},{2,3,4,4}}
   {{1,3},{1,1,2},{2,3,3}}
   {{1,3},{1,2,2},{2,3,3}}
   {{1,4},{1,5},{2,3,4,5}}
   {{2,3},{1,2,4},{3,4,4}}
   {{2,4},{1,2,3},{3,4,4}}
   {{2,4},{1,2,5},{3,4,5}}

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A317757, A318715, A318717.

Cf. A319752, A319759, A319762, A319764, A319765, A319779, A319781.

A319764 Number of non-isomorphic intersecting set systems of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 3, 8, 18

Offset: 0

Gus Wiseman, Sep 27 2018

A set system is a finite set of finite nonempty sets. It is intersecting if no two parts are disjoint. The weight of a set system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(6) = 1 through a(9) = 8 set systems:
6: {{1,2},{1,3},{2,3}}
7: {{1,3},{1,4},{2,3,4}}
8: {{1,2},{1,3,4},{2,3,4}}
   {{1,4},{1,5},{2,3,4,5}}
   {{2,4},{1,2,5},{3,4,5}}
9: {{1,3},{1,4,5},{2,3,4,5}}
   {{1,5},{1,6},{2,3,4,5,6}}
   {{2,5},{1,2,6},{3,4,5,6}}
   {{1,2,3},{2,4,5},{3,4,5}}
   {{1,3,5},{2,3,6},{4,5,6}}
   {{1,2},{1,3},{1,4},{2,3,4}}
   {{1,2},{1,3},{2,3},{1,2,3}}
   {{1,3},{1,4},{3,4},{2,3,4}}

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A317752, A317755, A317757, A318715, A318717.

Cf. A319752, A319759, A319762, A319763, A319765, A319779, A319781.

A328679 Heinz numbers of integer partitions with no two distinct parts relatively prime, but with no divisor in common to all of the parts.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 17719, 32768, 40807, 43381, 50431, 65536, 74269, 83143, 101543, 105703, 116143, 121307, 123469, 131072, 139919, 140699, 142883, 171613, 181831, 185803, 191479, 203557, 205813, 211381, 213239

Offset: 1

Gus Wiseman, Oct 30 2019

nonn

Equals the union A000079 and A328868.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
A partition with no two distinct parts relatively prime is said to be intersecting.

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      4: {1,1}
      8: {1,1,1}
     16: {1,1,1,1}
     32: {1,1,1,1,1}
     64: {1,1,1,1,1,1}
    128: {1,1,1,1,1,1,1}
    256: {1,1,1,1,1,1,1,1}
    512: {1,1,1,1,1,1,1,1,1}
   1024: {1,1,1,1,1,1,1,1,1,1}
   2048: {1,1,1,1,1,1,1,1,1,1,1}
   4096: {1,1,1,1,1,1,1,1,1,1,1,1}
   8192: {1,1,1,1,1,1,1,1,1,1,1,1,1}
  16384: {1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  17719: {6,10,15}
  32768: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  40807: {6,14,21}
  43381: {6,15,20}
  50431: {10,12,15}
  65536: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}

These are the Heinz numbers of the partitions counted by A328672.
Terms that are not powers of 2 are A328868.
The strict case is A318716.
The version without global relative primality is A328867.
A ranking using binary indices (instead of prime indices) is A326912.
The version for non-isomorphic multiset partitions is A319759.
The version for divisibility (instead of relative primality) is A328677.
Heinz numbers of relatively prime partitions are A289509.
Cf. A000837, A056239, A112798, A200976, A202425, A289509, A291166, A302796, A316476, A318715, A319752, A328336.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[10000],#==1||GCD@@primeMS[#]==1&&And[And@@(GCD[##]>1&)@@@Subsets[Union[primeMS[#]],{2}]]&]

cp's OEIS Frontend

A319760 Number of non-isomorphic intersecting strict multiset partitions (sets of multisets) of weight n.

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A328672 Number of integer partitions of n with relatively prime parts in which no two distinct parts are relatively prime.

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

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A328868 Heinz numbers of integer partitions with no two (not necessarily distinct) parts relatively prime, but with no divisor in common to all of the parts.

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A319762 Number of non-isomorphic intersecting set multipartitions (multisets of sets) of weight n with empty intersection.

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A319763 Number of non-isomorphic strict intersecting multiset partitions (sets of multisets) of weight n with empty intersection.

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A319764 Number of non-isomorphic intersecting set systems of weight n with empty intersection.

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A328679 Heinz numbers of integer partitions with no two distinct parts relatively prime, but with no divisor in common to all of the parts.

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