A335239 -id:A335239 - cp's OEIS frontend

A333228 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are pairwise coprime, where a singleton is not considered coprime unless it is (1).

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Gus Wiseman, May 28 2020

nonn

First differs from A291166 in lacking 69, which corresponds to the composition (4,2,1).
We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).
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:
   1: (1)          21: (2,2,1)        39: (3,1,1,1)
   3: (1,1)        22: (2,1,2)        41: (2,3,1)
   5: (2,1)        23: (2,1,1,1)      43: (2,2,1,1)
   6: (1,2)        24: (1,4)          44: (2,1,3)
   7: (1,1,1)      25: (1,3,1)        45: (2,1,2,1)
   9: (3,1)        26: (1,2,2)        46: (2,1,1,2)
  11: (2,1,1)      27: (1,2,1,1)      47: (2,1,1,1,1)
  12: (1,3)        28: (1,1,3)        48: (1,5)
  13: (1,2,1)      29: (1,1,2,1)      49: (1,4,1)
  14: (1,1,2)      30: (1,1,1,2)      50: (1,3,2)
  15: (1,1,1,1)    31: (1,1,1,1,1)    51: (1,3,1,1)
  17: (4,1)        33: (5,1)          52: (1,2,3)
  18: (3,2)        35: (4,1,1)        53: (1,2,2,1)
  19: (3,1,1)      37: (3,2,1)        54: (1,2,1,2)
  20: (2,3)        38: (3,1,2)        55: (1,2,1,1,1)

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

Pairwise coprime or singleton partitions are A051424.
Coprime or singleton sets are ranked by A087087.
The version for relatively prime instead of coprime appears to be A291166.
Numbers whose binary indices are pairwise coprime are A326675.
Coprime partitions are counted by A327516.
Not ignoring repeated parts gives A333227.
The complement is A335238.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Product is A124758.
- Reverse is A228351
- GCD is A326674.
- Heinz number is A333219.
- LCM is A333226.
- Number of distinct parts is A334028.
Cf. A007360, A048793, A101268, A233564, A272919, A291166, A302569, A335235, A335236, A335237, A335239.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,120],CoprimeQ@@Union[stc[#]]&]

A337462 Number of pairwise coprime compositions of n, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

1, 1, 1, 3, 6, 12, 21, 37, 62, 100, 159, 253, 402, 634, 983, 1491, 2224, 3280, 4813, 7043, 10270, 14888, 21415, 30585, 43400, 61204, 85747, 119295, 164834, 226422, 309663, 422301, 574826, 781236, 1060181, 1436367, 1942588, 2622078, 3531151, 4742315, 6348410

Offset: 0

Gus Wiseman, Sep 18 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(1) = 1 through a(5) = 12 compositions:
  (1)  (1,1)  (1,2)    (1,3)      (1,4)
              (2,1)    (3,1)      (2,3)
              (1,1,1)  (1,1,2)    (3,2)
                       (1,2,1)    (4,1)
                       (2,1,1)    (1,1,3)
                       (1,1,1,1)  (1,3,1)
                                  (3,1,1)
                                  (1,1,1,2)
                                  (1,1,2,1)
                                  (1,2,1,1)
                                  (2,1,1,1)
                                  (1,1,1,1,1)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500

A000740 counts the relatively prime instead of pairwise coprime version.
A101268 considers all singletons to be coprime, with strict case A337562.
A327516 is the unordered version.
A333227 ranks these compositions, with complement A335239.
A337461 counts these compositions of length 3.
A337561 is the strict case.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A178472 counts compositions with a common factor.
A305713 counts strict pairwise coprime partitions.
A328673 counts pairwise non-coprime partitions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A337667 counts pairwise non-coprime compositions.
Cf. A001523, A007360, A087087, A220377, A302569, A307719, A326675, A335235, A335238, A337664.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],#=={}||CoprimeQ@@#&]],{n,0,10}]

For n > 1, a(n) = A101268(n) - 1.

A335235 Numbers k such that the k-th composition in standard order (A066099) is pairwise coprime, where a singleton is always considered coprime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 35, 37, 38, 39, 41, 44, 47, 48, 49, 50, 51, 52, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 71, 72, 75, 77, 78, 79, 80, 83, 89, 92, 95, 96, 97

Offset: 1

Gus Wiseman, May 28 2020

nonn

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:
   1: (1)          20: (2,3)          48: (1,5)
   2: (2)          23: (2,1,1,1)      49: (1,4,1)
   3: (1,1)        24: (1,4)          50: (1,3,2)
   4: (3)          25: (1,3,1)        51: (1,3,1,1)
   5: (2,1)        27: (1,2,1,1)      52: (1,2,3)
   6: (1,2)        28: (1,1,3)        55: (1,2,1,1,1)
   7: (1,1,1)      29: (1,1,2,1)      56: (1,1,4)
   8: (4)          30: (1,1,1,2)      57: (1,1,3,1)
   9: (3,1)        31: (1,1,1,1,1)    59: (1,1,2,1,1)
  11: (2,1,1)      32: (6)            60: (1,1,1,3)
  12: (1,3)        33: (5,1)          61: (1,1,1,2,1)
  13: (1,2,1)      35: (4,1,1)        62: (1,1,1,1,2)
  14: (1,1,2)      37: (3,2,1)        63: (1,1,1,1,1,1)
  15: (1,1,1,1)    38: (3,1,2)        64: (7)
  16: (5)          39: (3,1,1,1)      65: (6,1)
  17: (4,1)        41: (2,3,1)        66: (5,2)
  18: (3,2)        44: (2,1,3)        67: (5,1,1)
  19: (3,1,1)      47: (2,1,1,1,1)    68: (4,3)

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

The version counting partitions is A051424, with strict case A007360.
The version for binary indices is A087087.
The version counting compositions is A101268.
The version for prime indices is A302569.
The case without singletons is A333227.
The complement is A335236.
Numbers whose binary indices are pairwise coprime are A326675.
Coprime partitions are counted by A327516.
All of the following pertain to compositions in standard order:
- Length is A000120.
- The parts are row k of A066099.
- Sum is A070939.
- Product is A124758.
- Reverse is A228351
- GCD is A326674.
- Heinz number is A333219.
- LCM is A333226.
Cf. A048793, A272919, A291166, A302569, A335236, A335237, A335238, A335239, A335240.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],Length[stc[#]]==1||CoprimeQ@@stc[#]&]

A335238 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are not pairwise coprime, where a singleton is not coprime unless it is (1).

Original entry on oeis.org

0, 2, 4, 8, 10, 16, 32, 34, 36, 40, 42, 64, 69, 70, 81, 88, 98, 104, 128, 130, 136, 138, 139, 141, 142, 160, 162, 163, 168, 170, 177, 184, 197, 198, 209, 216, 226, 232, 256, 260, 261, 262, 274, 276, 277, 278, 279, 282, 283, 285, 286, 288, 290, 292, 296, 321

Offset: 1

Gus Wiseman, May 28 2020

nonn

We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).

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: ()          88: (2,1,4)      177: (2,1,4,1)
    2: (2)         98: (1,4,2)      184: (2,1,1,4)
    4: (3)        104: (1,2,4)      197: (1,4,2,1)
    8: (4)        128: (8)          198: (1,4,1,2)
   10: (2,2)      130: (6,2)        209: (1,2,4,1)
   16: (5)        136: (4,4)        216: (1,2,1,4)
   32: (6)        138: (4,2,2)      226: (1,1,4,2)
   34: (4,2)      139: (4,2,1,1)    232: (1,1,2,4)
   36: (3,3)      141: (4,1,2,1)    256: (9)
   40: (2,4)      142: (4,1,1,2)    260: (6,3)
   42: (2,2,2)    160: (2,6)        261: (6,2,1)
   64: (7)        162: (2,4,2)      262: (6,1,2)
   69: (4,2,1)    163: (2,4,1,1)    274: (4,3,2)
   70: (4,1,2)    168: (2,2,4)      276: (4,2,3)
   81: (2,4,1)    170: (2,2,2,2)    277: (4,2,2,1)

The complement is A333228.
Not ignoring repeated parts gives A335239.
Singleton or pairwise coprime partitions are counted by A051424.
Singleton or pairwise coprime sets are ranked by A087087.
Coprime partitions are counted by A327516.
Non-coprime partitions are counted by A335240.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Product is A124758.
- Reverse is A228351
- GCD is A326674.
- Heinz number is A333219.
- LCM is A333226.
- Coprime compositions are A333227.
- Compositions whose distinct parts are coprime are A333228.
- Number of distinct parts is A334028.
Cf. A007360, A048793, A101268, A291166, A302569, A326675, A335235, A335236, A335237.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!CoprimeQ@@Union[stc[#]]&]

A335236 Numbers k such that the k-th composition in standard order (A066099) is not a singleton nor pairwise coprime.

Original entry on oeis.org

0, 10, 21, 22, 26, 34, 36, 40, 42, 43, 45, 46, 53, 54, 58, 69, 70, 73, 74, 76, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 98, 100, 104, 106, 107, 109, 110, 117, 118, 122, 130, 136, 138, 139, 141, 142, 146, 147, 148, 149, 150, 153, 154, 156, 160, 162, 163, 164

Offset: 1

Gus Wiseman, May 28 2020

nonn

These are compositions whose product is strictly greater than the LCM of their parts.

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: ()            74: (3,2,2)        109: (1,2,1,2,1)
   10: (2,2)         76: (3,1,3)        110: (1,2,1,1,2)
   21: (2,2,1)       81: (2,4,1)        117: (1,1,2,2,1)
   22: (2,1,2)       82: (2,3,2)        118: (1,1,2,1,2)
   26: (1,2,2)       84: (2,2,3)        122: (1,1,1,2,2)
   34: (4,2)         85: (2,2,2,1)      130: (6,2)
   36: (3,3)         86: (2,2,1,2)      136: (4,4)
   40: (2,4)         87: (2,2,1,1,1)    138: (4,2,2)
   42: (2,2,2)       88: (2,1,4)        139: (4,2,1,1)
   43: (2,2,1,1)     90: (2,1,2,2)      141: (4,1,2,1)
   45: (2,1,2,1)     91: (2,1,2,1,1)    142: (4,1,1,2)
   46: (2,1,1,2)     93: (2,1,1,2,1)    146: (3,3,2)
   53: (1,2,2,1)     94: (2,1,1,1,2)    147: (3,3,1,1)
   54: (1,2,1,2)     98: (1,4,2)        148: (3,2,3)
   58: (1,1,2,2)    100: (1,3,3)        149: (3,2,2,1)
   69: (4,2,1)      104: (1,2,4)        150: (3,2,1,2)
   70: (4,1,2)      106: (1,2,2,2)      153: (3,1,3,1)
   73: (3,3,1)      107: (1,2,2,1,1)    154: (3,1,2,2)

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

The version for prime indices is A316438.
The version for binary indices is A335237.
The complement is A335235.
The version with singletons allowed is A335239.
Binary indices are pairwise coprime or a singleton: A087087.
The version counting partitions is 1 + A335240.
All of the following pertain to compositions in standard order:
- Length is A000120.
- The parts are row k of A066099.
- Sum is A070939.
- Product is A124758.
- Reverse is A228351
- GCD is A326674.
- Heinz number is A333219.
- LCM is A333226.
Cf. A007360, A048793, A051424, A101268, A272919, A291166, A302569, A326675, A333227, A333228, A335238.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!(Length[stc[#]]==1||CoprimeQ@@stc[#])&]

A337664 Number of compositions of n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 30, 58, 111, 210, 396, 750, 1420, 2688, 5079, 9586, 18092, 34157, 64516, 121899, 230373, 435463, 823379, 1557421, 2946938, 5578111, 10561990, 20005129, 37902514, 71832373, 136173273, 258211603, 489738627, 929074448, 1762899110, 3345713034

Offset: 0

Gus Wiseman, Sep 21 2020

nonn

			The a(0) = 1 through a(5) = 16 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (211)   (122)
                        (1111)  (131)
                                (212)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..220

A304712 is the unordered version.
A337562 is the strict case.
A337602 is the length-3 case.
A337665 does not consider a singleton to be coprime unless it is (1).
A337695 ranks the complement of these compositions.
A000740 counts relatively prime compositions.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A337461 counts pairwise coprime length-3 compositions.
A337561 counts pairwise coprime strict compositions.
Cf. A007359, A007360, A087087, A302569, A304709, A307719, A335235, A335238, A335239, A337562, A337603, A337667.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@#||CoprimeQ@@Union[#]&]],{n,0,15}]

A335240 Number of integer partitions of n that are not pairwise coprime, where a singleton is not coprime unless it is (1).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 5, 6, 11, 16, 25, 34, 51, 69, 98, 134, 181, 238, 316, 410, 536, 691, 887, 1122, 1423, 1788, 2246, 2800, 3483, 4300, 5304, 6508, 7983, 9745, 11869, 14399, 17436, 21040, 25367, 30482, 36568, 43735, 52239, 62239, 74073, 87950, 104277, 123348

Offset: 0

Gus Wiseman, May 30 2020

nonn

We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).

These are also partitions that are a singleton or whose product is strictly greater than the LCM of their parts.

			The a(2) = 1 through a(9) = 16 partitions:
  (2)  (3)  (4)   (5)    (6)     (7)      (8)       (9)
            (22)  (221)  (33)    (322)    (44)      (63)
                         (42)    (331)    (62)      (333)
                         (222)   (421)    (332)     (432)
                         (2211)  (2221)   (422)     (441)
                                 (22111)  (2222)    (522)
                                          (3221)    (621)
                                          (3311)    (3222)
                                          (4211)    (3321)
                                          (22211)   (4221)
                                          (221111)  (22221)
                                                    (32211)
                                                    (33111)
                                                    (42111)
                                                    (222111)
                                                    (2211111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..750

The version for relatively prime instead of coprime is A018783.
The Heinz numbers of these partitions are the complement of A302696.
The complement is counted by A327516.
Singleton or pairwise coprime partitions are counted by A051424.
Singleton or pairwise coprime sets are ranked by A087087.
Numbers whose binary indices are pairwise coprime are A326675.
All of the following pertain to compositions in standard order (A066099):
- GCD is A326674.
- LCM is A333226.
- Coprime compositions are A333227.
- Compositions whose distinct parts are coprime are A333228.
- Non-coprime compositions are A335239.
Cf. A007360, A101268, A302569, A335235, A335236, A335237, A335238.

Table[Length[Select[IntegerPartitions[n],!CoprimeQ@@#&]],{n,0,30}]

A335237 Numbers whose binary indices are not a singleton nor pairwise coprime.

Original entry on oeis.org

0, 10, 11, 14, 15, 26, 27, 30, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 74, 75, 78, 79, 90, 91, 94, 95, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 114, 115, 116

Offset: 1

Gus Wiseman, May 28 2020

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.

			The sequence of terms together with their binary expansions and binary indices begins:
    0:       0 ~ {}
   10:    1010 ~ {2,4}
   11:    1011 ~ {1,2,4}
   14:    1110 ~ {2,3,4}
   15:    1111 ~ {1,2,3,4}
   26:   11010 ~ {2,4,5}
   27:   11011 ~ {1,2,4,5}
   30:   11110 ~ {2,3,4,5}
   31:   11111 ~ {1,2,3,4,5}
   34:  100010 ~ {2,6}
   35:  100011 ~ {1,2,6}
   36:  100100 ~ {3,6}
   37:  100101 ~ {1,3,6}
   38:  100110 ~ {2,3,6}
   39:  100111 ~ {1,2,3,6}
   40:  101000 ~ {4,6}
   41:  101001 ~ {1,4,6}
   42:  101010 ~ {2,4,6}
   43:  101011 ~ {1,2,4,6}
   44:  101100 ~ {3,4,6}

The version for prime indices is A316438.
The version for standard compositions is A335236.
Numbers whose binary indices are pairwise coprime or a singleton: A087087.
Non-coprime partitions are counted by A335240.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Product is A124758.
- Reverse is A228351
- GCD is A326674.
- Heinz number is A333219.
- LCM is A333226.
Cf. A007360, A048793, A051424, A101268, A291166, A302569, A326675, A333227, A333228, A335235, A335239.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],!(Length[bpe[#]]==1||CoprimeQ@@bpe[#])&]

Complement in A001477 of A326675 and A000079.

A335241 Numbers whose prime indices are not pairwise coprime, where a singleton is not coprime unless it is {1}.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 18, 19, 21, 23, 25, 27, 29, 31, 36, 37, 39, 41, 42, 43, 45, 47, 49, 50, 53, 54, 57, 59, 61, 63, 65, 67, 71, 72, 73, 75, 78, 79, 81, 83, 84, 87, 89, 90, 91, 97, 98, 99, 100, 101, 103, 105, 107, 108, 109, 111, 113, 114, 115, 117, 121

Offset: 1

Gus Wiseman, May 30 2020

nonn

We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).

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 sequence of terms together with their prime indices begins:
    1: {}          31: {11}          61: {18}
    3: {2}         36: {1,1,2,2}     63: {2,2,4}
    5: {3}         37: {12}          65: {3,6}
    7: {4}         39: {2,6}         67: {19}
    9: {2,2}       41: {13}          71: {20}
   11: {5}         42: {1,2,4}       72: {1,1,1,2,2}
   13: {6}         43: {14}          73: {21}
   17: {7}         45: {2,2,3}       75: {2,3,3}
   18: {1,2,2}     47: {15}          78: {1,2,6}
   19: {8}         49: {4,4}         79: {22}
   21: {2,4}       50: {1,3,3}       81: {2,2,2,2}
   23: {9}         53: {16}          83: {23}
   25: {3,3}       54: {1,2,2,2}     84: {1,1,2,4}
   27: {2,2,2}     57: {2,8}         87: {2,10}
   29: {10}        59: {17}          89: {24}

The complement is A302696.
The version for relatively prime instead of coprime is A318978.
The version for standard compositions is A335239.
These are the Heinz numbers of the partitions counted by A335240.
Singleton or pairwise coprime partitions are counted by A051424.
Singleton or pairwise coprime sets are ranked by A087087.
Primes and numbers with pairwise coprime prime indices are A302569.
Numbers whose binary indices are pairwise coprime are A326675.
Coprime standard composition numbers are A333227.
Cf. A007360, A101268, A326674, A327516, A333228, A335236, A335237, A335238.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!CoprimeQ@@primeMS[#]&]

A336620 Numbers that are not a product of elements of A304711.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 42, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 78, 79, 81, 83, 87, 89, 91, 97, 101, 103, 105, 107, 109, 111, 113, 114, 115, 117, 121, 125, 126, 127, 129, 130, 131, 133, 137, 139, 147, 149

Offset: 1

Gus Wiseman, Aug 02 2020

nonn

A304711 lists numbers whose distinct prime indices are pairwise coprime.

The first term divisible by 4 is a(421) = 1092.

			The sequence of terms together with their prime indices begins:
      3: {2}         39: {2,6}       78: {1,2,6}
      5: {3}         41: {13}        79: {22}
      7: {4}         42: {1,2,4}     81: {2,2,2,2}
      9: {2,2}       43: {14}        83: {23}
     11: {5}         47: {15}        87: {2,10}
     13: {6}         49: {4,4}       89: {24}
     17: {7}         53: {16}        91: {4,6}
     19: {8}         57: {2,8}       97: {25}
     21: {2,4}       59: {17}       101: {26}
     23: {9}         61: {18}       103: {27}
     25: {3,3}       63: {2,2,4}    105: {2,3,4}
     27: {2,2,2}     65: {3,6}      107: {28}
     29: {10}        67: {19}       109: {29}
     31: {11}        71: {20}       111: {2,12}
     37: {12}        73: {21}       113: {30}

Gus Wiseman, Sequences counting and encoding certain classes of multisets

A336426 is the version for superprimorials, with complement A181818.
A336497 is the version for superfactorials, with complement A336496.
A336735 is the complement.
A000837 counts relatively prime partitions, with strict case A007360.
A001055 counts factorizations.
A302696 lists numbers with coprime prime indices.
A304711 lists numbers with coprime distinct prime indices.
Cf. A007916, A112798, A302569, A327516, A333228, A335238, A335239, A335240, A335241, A336424, A336568, A336736.

nn=100;
dat=Select[Range[nn],CoprimeQ@@PrimePi/@First/@FactorInteger[#]&];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Select[Range[nn],facsusing[dat,#]=={}&]

cp's OEIS Frontend

A333228 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are pairwise coprime, where a singleton is not considered coprime unless it is (1).

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A337462 Number of pairwise coprime compositions of n, where a singleton is not considered coprime unless it is (1).

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A335235 Numbers k such that the k-th composition in standard order (A066099) is pairwise coprime, where a singleton is always considered coprime.

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A335238 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are not pairwise coprime, where a singleton is not coprime unless it is (1).

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A335236 Numbers k such that the k-th composition in standard order (A066099) is not a singleton nor pairwise coprime.

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A337664 Number of compositions of n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.

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A335240 Number of integer partitions of n that are not pairwise coprime, where a singleton is not coprime unless it is (1).

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A335237 Numbers whose binary indices are not a singleton nor pairwise coprime.

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