A305148 -id:A305148 - cp's OEIS frontend

A338331 Numbers whose set of distinct prime indices (A304038) is pairwise coprime, where a singleton is always considered coprime.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Gus Wiseman, Oct 31 2020

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.

Also Heinz numbers of partitions whose set of distinct parts is a singleton or pairwise coprime. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      1: {}          16: {1,1,1,1}     32: {1,1,1,1,1}
      2: {1}         17: {7}           33: {2,5}
      3: {2}         18: {1,2,2}       34: {1,7}
      4: {1,1}       19: {8}           35: {3,4}
      5: {3}         20: {1,1,3}       36: {1,1,2,2}
      6: {1,2}       22: {1,5}         37: {12}
      7: {4}         23: {9}           38: {1,8}
      8: {1,1,1}     24: {1,1,1,2}     40: {1,1,1,3}
      9: {2,2}       25: {3,3}         41: {13}
     10: {1,3}       26: {1,6}         43: {14}
     11: {5}         27: {2,2,2}       44: {1,1,5}
     12: {1,1,2}     28: {1,1,4}       45: {2,2,3}
     13: {6}         29: {10}          46: {1,9}
     14: {1,4}       30: {1,2,3}       47: {15}
     15: {2,3}       31: {11}          48: {1,1,1,1,2}

A302798 is the squarefree case.
A304709 counts partitions with pairwise coprime distinct parts, with ordered version A337665 and Heinz numbers A304711.
A304711 does not consider singletons relatively prime, except for (1).
A304712 counts the partitions with these Heinz numbers.
A316476 is the version for indivisibility instead of relative primality.
A328867 is the pairwise non-coprime instead of pairwise coprime version.
A337600 counts triples of this type, with ordered version A337602.
A338330 is the complement.
A000961 lists powers of primes.
A051424 counts pairwise coprime or singleton partitions.
A304038 gives the distinct prime indices of each positive integer.
A327516 counts pairwise coprime partitions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A000837, A047968, A056239, A112798, A289509, A302797, A305148, A318716, A318719, A337664, A337695.

Select[Range[100],#==1||PrimePowerQ[#]||CoprimeQ@@PrimePi/@First/@FactorInteger[#]&]

Equals A304711 \/ A000961.

A371445 Numbers whose distinct prime indices are binary carry-connected and have no binary containments.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 55, 59, 61, 64, 65, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 128, 131, 137, 139, 143, 145, 149, 151, 157, 163, 167, 169, 173, 179, 181

Offset: 1

Gus Wiseman, Mar 30 2024

nonn

Also Heinz numbers of binary carry-connected integer partitions whose distinct parts have no binary containments, counted by A371446.
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.
A binary carry of two positive integers is an overlap of binary indices. A multiset is said to be binary carry-connected iff the graph whose vertices are the elements and whose edges are binary carries is connected.
A binary containment is a containment of binary indices. For example, the numbers {3,5} have binary indices {{1,2},{1,3}}, so there is a binary carry but not a binary containment.

			The terms together with their prime indices begin:
     2: {1}            37: {12}              97: {25}
     3: {2}            41: {13}             101: {26}
     4: {1,1}          43: {14}             103: {27}
     5: {3}            47: {15}             107: {28}
     7: {4}            49: {4,4}            109: {29}
     8: {1,1,1}        53: {16}             113: {30}
     9: {2,2}          55: {3,5}            115: {3,9}
    11: {5}            59: {17}             121: {5,5}
    13: {6}            61: {18}             125: {3,3,3}
    16: {1,1,1,1}      64: {1,1,1,1,1,1}    127: {31}
    17: {7}            65: {3,6}            128: {1,1,1,1,1,1,1}
    19: {8}            67: {19}             131: {32}
    23: {9}            71: {20}             137: {33}
    25: {3,3}          73: {21}             139: {34}
    27: {2,2,2}        79: {22}             143: {5,6}
    29: {10}           81: {2,2,2,2}        145: {3,10}
    31: {11}           83: {23}             149: {35}
    32: {1,1,1,1,1}    89: {24}             151: {36}

Contains all powers of primes A000961 except 1.
Case of A325118 (counted by A325098) without binary containments.
For binary indices of binary indices we have A326750 = A326704 /\ A326749.
For prime indices of prime indices we have A329559 = A305078 /\ A316476.
An opposite version is A371294 = A087086 /\ A371291.
Partitions of this type are counted by A371446.
Carry-connected case of A371455 (counted by A325109).
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.
A326964 counts connected set-systems, covering A323818.
Cf. A019565, A056239, A112798, A304713, A304716, A305079, A305148, A325097, A325105, A325107, A325119, A371452.

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}]]]];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Select[Range[100],stableQ[bpe/@prix[#],SubsetQ] && Length[csm[bpe/@prix[#]]]==1&]

Intersection of A371455 and A325118.

A319728 Number of strict T_0 integer partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 6, 8, 9, 10, 14, 16, 19, 25, 31, 34, 41, 49, 59, 72, 81, 94, 113, 133, 152, 179, 209, 239, 273, 315, 366, 422, 478, 548, 627, 711, 812, 926, 1051, 1185, 1340, 1514, 1718, 1945, 2179, 2444, 2757, 3095, 3465, 3892, 4362, 4865, 5427, 6068

Offset: 0

Gus Wiseman, Sep 26 2018

nonn

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. For an integer partition the T_0 condition means the dual of the multiset partition obtained by factoring each part into prime numbers is strict (no repeated blocks).

			The a(11) = 10 integer partitions are (11), (7,4), (8,3), (9,2), (5,4,2), (6,3,2), (6,4,1), (7,3,1), (8,2,1), (5,3,2,1). Missing from this list are (6,5) and (10,1).

Cf. A000009, A000041, A001970, A007716, A059201, A305148, A316983, A319558, A319564, A319616.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}]
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@dual[primeMS/@#]&]],{n,60}]

A371446 Number of carry-connected integer partitions whose distinct parts have no binary containments.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 2, 5, 4, 4, 4, 8, 4, 7, 7, 12, 10, 14, 12, 15, 19, 19, 21, 32, 27, 33, 40, 46, 47, 61, 52, 75, 89, 95, 104, 129, 129, 149, 176, 188, 208, 249, 257, 296, 341, 373, 394, 476, 496, 552

Offset: 0

Gus Wiseman, Apr 02 2024

These partitions are ranked by A371445.
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.
A binary carry of two positive integers is an overlap of binary indices. An integer partition is binary carry-connected iff the graph with one vertex for each part and edges corresponding to binary carries is connected.
A binary containment is a containment of binary indices. For example, the numbers {3,5} have binary indices {{1,2},{1,3}}, so there is a binary carry but not a binary containment.

			The a(12) = 8 through a(14) = 7 partitions:
  (12)             (13)                         (14)
  (6,6)            (10,3)                       (7,7)
  (9,3)            (5,5,3)                      (9,5)
  (4,4,4)          (1,1,1,1,1,1,1,1,1,1,1,1,1)  (6,5,3)
  (6,3,3)                                       (5,3,3,3)
  (3,3,3,3)                                     (2,2,2,2,2,2,2)
  (2,2,2,2,2,2)                                 (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)

The first condition (carry-connected) is A325098.
The second condition (stable) is A325109.
Ranks for binary indices of binary indices are A326750 = A326704 /\ A326749.
Ranks for prime indices of prime indices are A329559 = A305078 /\ A316476.
Ranks for prime indices of binary indices are A371294 = A087086 /\ A371291.
Ranks for binary indices of prime indices are A371445 = A325118 /\ A371455.
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A326964 counts connected set-systems, covering A323818.
Cf. A019565, A304713, A304716, A305079, A305148, A325097, A325105, A325107, A325119, A371452.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]},If[c=={},s, csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n], stableQ[bix/@Union[#],SubsetQ]&&Length[csm[bix/@#]]<=1&]],{n,0,30}]

A323053 Number of integer partitions of n with no 1's such that no part is a power of any other (unequal) part.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 15, 19, 25, 30, 38, 47, 58, 71, 87, 106, 131, 156, 190, 228, 275, 328, 394, 468, 556, 661, 784, 923, 1089, 1283, 1507, 1766, 2068, 2416, 2821, 3284, 3822, 4438, 5148, 5961, 6898, 7968, 9195, 10593, 12198, 14019, 16102, 18472

Offset: 0

Gus Wiseman, Jan 04 2019

nonn

			The a(2) = 1 through a(11) = 12 integer partitions (A = 10, B = 11):
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)     (A)      (B)
            (22)  (32)  (33)   (43)   (44)    (54)    (55)     (65)
                        (222)  (52)   (53)    (63)    (64)     (74)
                               (322)  (62)    (72)    (73)     (83)
                                      (332)   (333)   (433)    (92)
                                      (2222)  (522)   (532)    (443)
                                              (3222)  (622)    (533)
                                                      (3322)   (632)
                                                      (22222)  (722)
                                                               (3332)
                                                               (5222)
                                                               (32222)

Cf. A001597, A002865, A007916, A052410, A101417, A102430, A108917, A305148, A305630, A305631, A321346, A323093.

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

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}]

A371179 Positive integers with fewer distinct prime factors (A001221) than distinct divisors of prime indices (A370820).

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 14, 15, 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 33, 35, 37, 38, 39, 41, 43, 45, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 63, 65, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99, 101

Offset: 1

Gus Wiseman, Mar 19 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.

			The terms together with their prime indices begin:
     3: {2}        28: {1,1,4}    52: {1,1,6}      74: {1,12}
     5: {3}        29: {10}       53: {16}         75: {2,3,3}
     7: {4}        31: {11}       55: {3,5}        76: {1,1,8}
     9: {2,2}      33: {2,5}      56: {1,1,1,4}    77: {4,5}
    11: {5}        35: {3,4}      57: {2,8}        78: {1,2,6}
    13: {6}        37: {12}       58: {1,10}       79: {22}
    14: {1,4}      38: {1,8}      59: {17}         81: {2,2,2,2}
    15: {2,3}      39: {2,6}      61: {18}         83: {23}
    17: {7}        41: {13}       63: {2,2,4}      85: {3,7}
    19: {8}        43: {14}       65: {3,6}        86: {1,14}
    21: {2,4}      45: {2,2,3}    67: {19}         87: {2,10}
    23: {9}        46: {1,9}      69: {2,9}        89: {24}
    25: {3,3}      47: {15}       70: {1,3,4}      91: {4,6}
    26: {1,6}      49: {4,4}      71: {20}         92: {1,1,9}
    27: {2,2,2}    51: {2,7}      73: {21}         93: {2,11}

The LHS is A001221, distinct case of A001222.
The RHS is A370820, for prime factors A303975.
Partitions of this type are counted by A371132, strict A371180.
Counting all prime indices on the LHS gives A371168, counted by A371173.
The complement is A371177, counted by A371178, strict A371128.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
A305148 counts pairwise indivisible (stable) partitions, ranks A316476.
Cf. A003963, A239312, A285573, A355529, A370802, A370803, A371130, A371165, A371171, A371172.

Mathematica
```
Select[Range[100],PrimeNu[#]
				
```

A001221(a(n)) < A370820(a(n)).

A371455 Numbers k such that if we take the binary indices of each prime index of k we get an antichain of sets.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 35, 36, 37, 38, 41, 42, 43, 47, 48, 49, 52, 53, 54, 55, 56, 57, 58, 59, 61, 63, 64, 65, 67, 69, 71, 72, 73, 74, 76, 79, 81, 83, 84, 86, 89, 95, 96, 97, 98, 99

Offset: 1

Gus Wiseman, Apr 01 2024

nonn

In an antichain of sets, no edge is a proper subset of any other.

			The prime indices of 65 are {3,6} with binary indices {{1,2},{2,3}} so 65 is in the sequence.
The prime indices of 255 are {2,3,7} with binary indices {{2},{1,2},{1,2,3}} so 255 is not in the sequence.

Jakub Buczak, Table of n, a(n) for n = 1..10000

Contains all powers of primes A000961.
An opposite version is A087086, carry-connected case A371294.
For prime indices of prime indices we have A316476, carry-connected A329559.
These antichains are counted by A325109.
For binary indices of binary indices we have A326704, carry-conn. A326750.
The carry-connected case is A371445, counted by A371446.
A048143 counts connected antichains of sets.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A050320 counts set multipartitions of prime indices, see also A318360.
A070939 gives length of binary expansion.
A089259 counts set multipartitions of integer partitions.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A116540 counts normal set multipartitions.
A302478 ranks set multipartitions, cf. A073576.
A325118 ranks carry-connected partitions, counted by A325098.
A371451 counts carry-connected components of binary indices.
Cf. A019565, A050326, A304713, A305148, A325097, A325107, A371291, A371452.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
bix[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[100],stableQ[bix/@prix[#],SubsetQ]&]

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

Original entry on oeis.org

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}]

A328675 Number of integer partitions of n with no two distinct consecutive parts divisible.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 9, 13, 13, 22, 23, 30, 36, 50, 54, 77, 85, 113, 135, 170, 194, 256, 303, 369, 440, 545, 640, 792, 931, 1132, 1347, 1616, 1909, 2295, 2712, 3225, 3799, 4519, 5310, 6278, 7365, 8675, 10170, 11928, 13940, 16314, 19046, 22223, 25856

Offset: 0

Gus Wiseman, Oct 29 2019

nonn

			The a(1) = 1 through a(10) = 9 partitions (A = 10).
  1  2   3    4     5      6       7        8         9          A
     11  111  22    32     33      43       44        54         55
              1111  11111  222     52       53        72         64
                           111111  322      332       333        73
                                   1111111  2222      432        433
                                            11111111  522        532
                                                      3222       3322
                                                      111111111  22222
                                                                 1111111111

The Heinz numbers of these partitions are given by A328674.
The case involving all consecutive parts (not just distinct) is A328171.
The version for relative primality instead of divisibility is A328187.
Partitions with all consecutive parts divisible are A003238.
Compositions without consecutive divisibilities are A328460.
Cf. A305148, A316476, A318726, A328172, A328508, A328593, A328598, A328603.

Table[Length[Select[IntegerPartitions[n],!MatchQ[Union[#],{_,x_,y_,_}/;Divisible[y,x]]&]],{n,0,30}]

cp's OEIS Frontend

A338331 Numbers whose set of distinct prime indices (A304038) is pairwise coprime, where a singleton is always considered coprime.

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A371445 Numbers whose distinct prime indices are binary carry-connected and have no binary containments.

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A319728 Number of strict T_0 integer partitions of n.

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A371446 Number of carry-connected integer partitions whose distinct parts have no binary containments.

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A323053 Number of integer partitions of n with no 1's such that no part is a power of any other (unequal) part.

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

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A371179 Positive integers with fewer distinct prime factors (A001221) than distinct divisors of prime indices (A370820).

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A371455 Numbers k such that if we take the binary indices of each prime index of k we get an antichain of sets.

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

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