A328671 -id:A328671 - cp's OEIS frontend

A087086 Primitive sets of integers, each subset mapped onto a unique binary integer, values here shown in decimal.

0, 1, 2, 4, 6, 8, 12, 16, 18, 20, 22, 24, 28, 32, 40, 48, 56, 64, 66, 68, 70, 72, 76, 80, 82, 84, 86, 88, 92, 96, 104, 112, 120, 128, 132, 144, 148, 160, 176, 192, 196, 208, 212, 224, 240, 256, 258, 264, 272, 274, 280, 288, 296, 304, 312, 320, 322, 328, 336, 338, 344

Offset: 0

Alan Sutcliffe (alansut(AT)ntlworld.com), Aug 14 2003

A primitive set of integers has no pair of elements one of which divides the other. Each element i in a subset contributes 2^(i-1) to the binary value for that subset. The integers missing from the sequence correspond to nonprimitive subsets.

			a(10)=22 since the 10th primitive set counting from 0 is {5,3,2}, which maps onto 10110 binary = 22 decimal.
From _Gus Wiseman_, Oct 31 2019: (Start)
The sequence of terms together with their binary expansions and binary indices begins:
   0:       0 ~ {}
   1:       1 ~ {1}
   2:      10 ~ {2}
   4:     100 ~ {3}
   6:     110 ~ {2,3}
   8:    1000 ~ {4}
  12:    1100 ~ {3,4}
  16:   10000 ~ {5}
  18:   10010 ~ {2,5}
  20:   10100 ~ {3,5}
  22:   10110 ~ {2,3,5}
  24:   11000 ~ {4,5}
  28:   11100 ~ {3,4,5}
(End)

Alan Sutcliffe, Divisors and Common Factors in Sets of Integers, awaiting publication

A051026 gives the number of primitive subsets of the integers 1 to n.
The version for prime indices (rather than binary indices) is A316476.
The relatively prime case is A328671.
Partitions with no consecutive divisible parts are A328171.
Compositions without consecutive divisible parts are A328460.
A ranking of antichains is A326704.
Cf. A000120, A048793, A070939, A285572, A285573, A303362, A304713, A305148, A328593.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[0,100],stableQ[Join@@Position[Reverse[IntegerDigits[#,2]],1],Divisible]&] (* Gus Wiseman, Oct 31 2019 *)

A328677 Numbers whose distinct prime indices are relatively prime and pairwise indivisible.

Original entry on oeis.org

2, 4, 8, 15, 16, 32, 33, 35, 45, 51, 55, 64, 69, 75, 77, 85, 93, 95, 99, 119, 123, 128, 135, 141, 143, 145, 153, 155, 161, 165, 175, 177, 187, 201, 205, 207, 209, 215, 217, 219, 221, 225, 245, 249, 253, 255, 256, 265, 275, 279, 287, 291, 295, 297, 309, 323

Offset: 1

Gus Wiseman, Oct 30 2019

nonn

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. Stable numbers are listed in A316476.

			The sequence of terms together with their prime indices begins:
    2: {1}
    4: {1,1}
    8: {1,1,1}
   15: {2,3}
   16: {1,1,1,1}
   32: {1,1,1,1,1}
   33: {2,5}
   35: {3,4}
   45: {2,2,3}
   51: {2,7}
   55: {3,5}
   64: {1,1,1,1,1,1}
   69: {2,9}
   75: {2,3,3}
   77: {4,5}
   85: {3,7}
   93: {2,11}
   95: {3,8}
   99: {2,2,5}
  119: {4,7}

These are the Heinz numbers of the partitions counted by A328676.
Numbers whose prime indices are relatively prime are A289509.
Partitions whose distinct parts are pairwise indivisible are A305148.
The version for binary indices (instead of prime indices) is A328671.
Cf. A000837, A056239, A112798, A285573, A289508, A303362, A304713, A327393, A328460.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[100],GCD@@primeMS[#]==1&&stableQ[primeMS[#],Divisible]&]

Intersection of A316476 and A289509.

A371294 Numbers whose binary indices are connected and pairwise indivisible, where two numbers are connected iff they have a common factor. A hybrid ranking sequence for connected antichains of multisets.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 40, 64, 128, 160, 256, 288, 296, 416, 512, 520, 544, 552, 640, 672, 800, 808, 928, 1024, 2048, 2176, 2304, 2432, 2560, 2688, 2816, 2944, 4096, 8192, 8200, 8224, 8232, 8320, 8352, 8480, 8488, 8608, 8704, 8712, 8736, 8744, 8832, 8864, 8992

Offset: 1

Gus Wiseman, Mar 28 2024

nonn

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.

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.

			The terms together with their prime indices of binary indices begin:
    1: {{}}
    2: {{1}}
    4: {{2}}
    8: {{1,1}}
   16: {{3}}
   32: {{1,2}}
   40: {{1,1},{1,2}}
   64: {{4}}
  128: {{1,1,1}}
  160: {{1,2},{1,1,1}}
  256: {{2,2}}
  288: {{1,2},{2,2}}
  296: {{1,1},{1,2},{2,2}}
  416: {{1,2},{1,1,1},{2,2}}
  512: {{1,3}}
  520: {{1,1},{1,3}}
  544: {{1,2},{1,3}}
  552: {{1,1},{1,2},{1,3}}
  640: {{1,1,1},{1,3}}
  672: {{1,2},{1,1,1},{1,3}}
  800: {{1,2},{2,2},{1,3}}
  808: {{1,1},{1,2},{2,2},{1,3}}
  928: {{1,2},{1,1,1},{2,2},{1,3}}

Connected case of A087086, relatively prime A328671.
For binary indices of binary indices we have A326750, non-primitive A326749.
For prime indices of prime indices we have A329559, non-primitive A305078.
Primitive case of A371291 = positions of ones in A371452.
For binary indices of prime indices we have A371445, non-primitive A325118.
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326964 counts connected set-systems, covering A323818.
Cf. A001222, A051026, A285572, A303362, A304713, A305079, A316476, A319496, A319719, A326704, A371446.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],stableQ[bpe[#],Divisible]&&connectedQ[prix/@bpe[#]]&]

Intersection of A087086 and A371291.

A328676 Number of relatively prime integer partitions of n whose distinct parts are pairwise indivisible.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 3, 5, 5, 11, 7, 16, 14, 18, 22, 34, 30, 47, 45, 59, 66, 89, 90, 118, 125, 159, 169, 218, 225, 289, 304, 369, 400, 486, 520, 636, 680, 806, 873, 1051, 1105, 1333, 1424, 1664, 1803, 2122, 2253, 2659, 2841, 3283, 3560, 4118, 4388, 5096

Offset: 1

Gus Wiseman, Oct 29 2019

nonn

			The a(4) = 1 through a(11) = 11 partitions:
  1111  32     111111  43       53        54         73          65
        11111          52       332       72         433         74
                       322      11111111  522        532         83
                       1111111            3222       3322        92
                                          111111111  1111111111  443
                                                                 533
                                                                 722
                                                                 3332
                                                                 5222
                                                                 32222
                                                                 11111111111

The Heinz numbers of these partitions are given by A328677.
The strict case is A328678.
The binary index version is A328671.
Relatively prime partitions are A000837.
Partitions whose distinct parts are pairwise indivisible are A305148.
Cf. A285573, A289509, A303362, A316476, A326912, A327393, A328171.

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

A329560 BII-numbers of antichains of sets with empty intersection.

Original entry on oeis.org

0, 3, 9, 10, 11, 12, 18, 33, 52, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 148, 160, 161, 164, 176, 180, 192, 258, 264, 266, 268, 274, 288, 292, 304, 308, 513, 520, 521, 524, 528, 532, 545, 560, 564, 772, 776, 780, 784, 788, 800, 804, 816, 820, 832

Offset: 1

Gus Wiseman, Nov 28 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 set-system (finite set of finite nonempty sets of positive integers) 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.
A set-system is an antichain if no edge is a proper subset of any other.
Empty intersection means there is no vertex in common to all the edges

			The sequence of terms together with their binary expansions and corresponding set-systems begins:
    0:          0 ~ {}
    3:         11 ~ {{1},{2}}
    9:       1001 ~ {{1},{3}}
   10:       1010 ~ {{2},{3}}
   11:       1011 ~ {{1},{2},{3}}
   12:       1100 ~ {{1,2},{3}}
   18:      10010 ~ {{2},{1,3}}
   33:     100001 ~ {{1},{2,3}}
   52:     110100 ~ {{1,2},{1,3},{2,3}}
  129:   10000001 ~ {{1},{4}}
  130:   10000010 ~ {{2},{4}}
  131:   10000011 ~ {{1},{2},{4}}
  132:   10000100 ~ {{1,2},{4}}
  136:   10001000 ~ {{3},{4}}
  137:   10001001 ~ {{1},{3},{4}}
  138:   10001010 ~ {{2},{3},{4}}
  139:   10001011 ~ {{2},{3},{4}}
  140:   10001100 ~ {{1,2},{3},{4}}
  144:   10010000 ~ {{1,3},{4}}
  146:   10010010 ~ {{2},{1,3},{4}}
  148:   10010100 ~ {{1,2},{1,3},{4}}

Intersection of A326911 and A326704.
BII-numbers of intersecting set-systems with empty intersecting are A326912.
Cf. A000120, A048793, A070939, A326031, A326701, A328671, A329561, A329626, A329628, A329661.

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],#==0||Intersection@@bpe/@bpe[#]=={}&&stableQ[bpe/@bpe[#],SubsetQ]&]

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.

A329366 Numbers whose distinct prime indices are pairwise indivisible (stable) and pairwise non-relatively prime (intersecting).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197

Offset: 1

Gus Wiseman, Nov 12 2019

nonn

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 Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A partition with no two distinct parts divisible is said to be stable, and a partition with no two distinct parts relatively prime is said to be intersecting, so these are Heinz numbers of stable intersecting partitions.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   37: {12}

Intersection of A316476 and A328867.
Heinz numbers of the partitions counted by A328871.
Replacing "intersecting" with "relatively prime" gives A328677.
Cf. A056239, A112798, A285573, A289509, A303362, A304713, A327393, A328671.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[100],stableQ[Union[primeMS[#]],GCD[#1,#2]==1&]&&stableQ[Union[primeMS[#]],Divisible]&]

A328871 Number of integer partitions of n whose distinct parts are pairwise indivisible (stable) and pairwise non-relatively prime (intersecting).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 2, 4, 3, 5, 2, 6, 2, 7, 5, 7, 2, 10, 2, 11, 7, 14, 2, 16, 4, 19, 8, 22, 2, 30, 3, 29, 14, 37, 8, 48, 4, 50, 19, 59, 5, 82, 4, 81, 28, 93, 8, 128, 9, 128, 38, 147, 8, 199, 19, 196, 52, 223, 12, 308

Offset: 0

Gus Wiseman, Nov 12 2019

nonn

A partition with no two distinct parts divisible is said to be stable, and a partition with no two distinct parts relatively prime is said to be intersecting, so these are just stable intersecting partitions.

			The a(1) = 1 through a(10) = 5 partitions (A = 10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              1111         222              2222      111111111  64
                           111111           11111111             22222
                                                                 1111111111

The Heinz numbers of these partitions are A329366.
Replacing "intersecting" with "relatively prime" gives A328676.
Stable partitions are A305148.
Intersecting partitions are A328673.
Cf. A000837, A285573, A303362, A305148, A316476, A328671, A328677.

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

cp's OEIS Frontend

A087086 Primitive sets of integers, each subset mapped onto a unique binary integer, values here shown in decimal.

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A328677 Numbers whose distinct prime indices are relatively prime and pairwise indivisible.

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A371294 Numbers whose binary indices are connected and pairwise indivisible, where two numbers are connected iff they have a common factor. A hybrid ranking sequence for connected antichains of multisets.

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A328676 Number of relatively prime integer partitions of n whose distinct parts are pairwise indivisible.

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A329560 BII-numbers of antichains of sets with empty intersection.

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

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A329366 Numbers whose distinct prime indices are pairwise indivisible (stable) and pairwise non-relatively prime (intersecting).

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A328871 Number of integer partitions of n whose distinct parts are pairwise indivisible (stable) and pairwise non-relatively prime (intersecting).

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