A337561 -id:A337561 - cp's OEIS frontend

A337602 Number of ordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.

0, 0, 0, 1, 3, 6, 10, 9, 18, 16, 24, 21, 43, 24, 51, 31, 54, 42, 94, 45, 102, 55, 99, 69, 163, 66, 150, 88, 168, 96, 265, 93, 228, 121, 246, 126, 337, 132, 315, 169, 342, 162, 487, 165, 420, 217, 411, 213, 619, 207, 558, 259, 540, 258, 784, 264, 654, 325, 660

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

			The a(3) = 1 through a(8) = 18 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)  (1,1,6)
           (1,2,1)  (1,2,2)  (1,2,3)  (1,3,3)  (1,2,5)
           (2,1,1)  (1,3,1)  (1,3,2)  (1,5,1)  (1,3,4)
                    (2,1,2)  (1,4,1)  (2,2,3)  (1,4,3)
                    (2,2,1)  (2,1,3)  (2,3,2)  (1,5,2)
                    (3,1,1)  (2,2,2)  (3,1,3)  (1,6,1)
                             (2,3,1)  (3,2,2)  (2,1,5)
                             (3,1,2)  (3,3,1)  (2,3,3)
                             (3,2,1)  (5,1,1)  (2,5,1)
                             (4,1,1)           (3,1,4)
                                               (3,2,3)
                                               (3,3,2)
                                               (3,4,1)
                                               (4,1,3)
                                               (4,3,1)
                                               (5,1,2)
                                               (5,2,1)
                                               (6,1,1)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

The complement in A014311 of A337695 ranks these compositions.
A220377*6 is the strict case.
A337600 is the unordered version.
A337603 does not consider a singleton to be coprime unless it is (1).
A337664 counts these compositions of any length.
A000740 counts relatively prime compositions.
A337561 counts pairwise coprime strict compositions.
A000217 counts 3-part compositions.
A001399/A069905/A211540 count 3-part partitions.
A023023 counts relatively prime 3-part partitions.
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 3-part compositions.
Cf. A000212, A007359, A087087, A284825, A302696, A304709, A304712, A307719, A328673, A335235, A335238, A337483, A337562, A337601.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],SameQ@@#||CoprimeQ@@Union[#]&]],{n,0,100}]

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}]

A337984 Heinz numbers of pairwise coprime integer partitions with no 1's, where a singleton is not considered coprime.

Original entry on oeis.org

15, 33, 35, 51, 55, 69, 77, 85, 93, 95, 119, 123, 141, 143, 145, 155, 161, 165, 177, 187, 201, 205, 209, 215, 217, 219, 221, 249, 253, 255, 265, 287, 291, 295, 309, 323, 327, 329, 335, 341, 355, 381, 385, 391, 395, 403, 407, 411, 413, 415, 437, 447, 451, 465

Offset: 1

Gus Wiseman, Oct 22 2020

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
     15: {2,3}     155: {3,11}     265: {3,16}
     33: {2,5}     161: {4,9}      287: {4,13}
     35: {3,4}     165: {2,3,5}    291: {2,25}
     51: {2,7}     177: {2,17}     295: {3,17}
     55: {3,5}     187: {5,7}      309: {2,27}
     69: {2,9}     201: {2,19}     323: {7,8}
     77: {4,5}     205: {3,13}     327: {2,29}
     85: {3,7}     209: {5,8}      329: {4,15}
     93: {2,11}    215: {3,14}     335: {3,19}
     95: {3,8}     217: {4,11}     341: {5,11}
    119: {4,7}     219: {2,21}     355: {3,20}
    123: {2,13}    221: {6,7}      381: {2,31}
    141: {2,15}    249: {2,23}     385: {3,4,5}
    143: {5,6}     253: {5,9}      391: {7,9}
    145: {3,10}    255: {2,3,7}    395: {3,22}

A005117 is a superset.
A337485 counts these partitions.
A302568 considers singletons to be coprime.
A304711 allows 1's, with squarefree version A302797.
A337694 is the pairwise non-coprime instead of pairwise coprime version.
A007359 counts partitions into singleton or pairwise coprime parts with no 1's
A101268 counts pairwise coprime or singleton compositions, ranked by A335235.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions, ranked by A302696.
A337462 counts pairwise coprime compositions, ranked by A333227.
A337561 counts pairwise coprime strict compositions.
A337665 counts compositions whose distinct parts are pairwise coprime, ranked by A333228.
A337667 counts pairwise non-coprime compositions, ranked by A337666.
A337697 counts pairwise coprime compositions with no 1's.
A337983 counts pairwise non-coprime strict compositions, with unordered version A318717 ranked by A318719.
Cf. A051424, A056239, A087087, A112798, A200976, A220377, A302569, A303140, A303282, A328673, A328867.

Select[Range[1,100,2],SquareFreeQ[#]&&CoprimeQ@@PrimePi/@First/@FactorInteger[#]&]

Equals A302568\A000040.

A337697 Number of pairwise coprime compositions of n with no 1's, where a singleton is not considered coprime.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 4, 2, 4, 8, 8, 14, 10, 16, 12, 30, 38, 46, 46, 48, 52, 62, 152, 96, 156, 112, 190, 256, 338, 420, 394, 326, 402, 734, 622, 1150, 802, 946, 898, 1730, 1946, 2524, 2200, 2328, 2308, 3356, 5816, 4772, 5350, 4890, 6282, 6316, 12092, 8902

Offset: 0

Gus Wiseman, Oct 06 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. These compositions must be strict.

			The a(5) = 2 through a(12) = 14 compositions (empty column indicated by dot):
  (2,3)  .  (2,5)  (3,5)  (2,7)  (3,7)    (2,9)  (5,7)
  (3,2)     (3,4)  (5,3)  (4,5)  (7,3)    (3,8)  (7,5)
            (4,3)         (5,4)  (2,3,5)  (4,7)  (2,3,7)
            (5,2)         (7,2)  (2,5,3)  (5,6)  (2,7,3)
                                 (3,2,5)  (6,5)  (3,2,7)
                                 (3,5,2)  (7,4)  (3,4,5)
                                 (5,2,3)  (8,3)  (3,5,4)
                                 (5,3,2)  (9,2)  (3,7,2)
                                                 (4,3,5)
                                                 (4,5,3)
                                                 (5,3,4)
                                                 (5,4,3)
                                                 (7,2,3)
                                                 (7,3,2)

A022340 intersected with A333227 is a ranking sequence (using standard compositions A066099) for these compositions.
A212804 does not require coprimality, with unordered version A002865.
A337450 is the relatively prime instead of pairwise coprime version, with strict case A337451 and unordered version A302698.
A337462 allows 1's, with strict case A337561 (or A101268 with singletons), unordered version A327516 with Heinz numbers A302696, and 3-part case A337461.
A337485 is the unordered version (or A007359 with singletons considered coprime), with Heinz numbers A337984.
A337563 is the case of unordered triples.
Cf. A078374, A178472, A302568, A302697, A305713, A307719, A332004, A337562.

Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],!MemberQ[#,1]&&CoprimeQ@@#&]],{n,0,30}]

For n > 1, the version where singletons are considered coprime is a(n) + 1.

A337695 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are not pairwise coprime, where a singleton is always considered coprime.

Original entry on oeis.org

34, 40, 69, 70, 81, 88, 98, 104, 130, 138, 139, 141, 142, 160, 162, 163, 168, 177, 184, 197, 198, 209, 216, 226, 232, 260, 261, 262, 274, 276, 277, 278, 279, 282, 283, 285, 286, 288, 290, 296, 321, 324, 325, 326, 327, 328, 337, 344, 352, 354, 355, 360, 369

Offset: 1

Gus Wiseman, Sep 22 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:
     34: (4,2)        163: (2,4,1,1)    277: (4,2,2,1)
     40: (2,4)        168: (2,2,4)      278: (4,2,1,2)
     69: (4,2,1)      177: (2,1,4,1)    279: (4,2,1,1,1)
     70: (4,1,2)      184: (2,1,1,4)    282: (4,1,2,2)
     81: (2,4,1)      197: (1,4,2,1)    283: (4,1,2,1,1)
     88: (2,1,4)      198: (1,4,1,2)    285: (4,1,1,2,1)
     98: (1,4,2)      209: (1,2,4,1)    286: (4,1,1,1,2)
    104: (1,2,4)      216: (1,2,1,4)    288: (3,6)
    130: (6,2)        226: (1,1,4,2)    290: (3,4,2)
    138: (4,2,2)      232: (1,1,2,4)    296: (3,2,4)
    139: (4,2,1,1)    260: (6,3)        321: (2,6,1)
    141: (4,1,2,1)    261: (6,2,1)      324: (2,4,3)
    142: (4,1,1,2)    262: (6,1,2)      325: (2,4,2,1)
    160: (2,6)        274: (4,3,2)      326: (2,4,1,2)
    162: (2,4,2)      276: (4,2,3)      327: (2,4,1,1,1)

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

A304712 counts the complement, with ordered version A337664.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A335238 does not consider a singleton coprime unless it is (1).
A337600 counts 3-part partitions in the complement.
A000740 counts relatively prime compositions.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A337461 counts pairwise coprime 3-part compositions.
A337561 counts pairwise coprime strict compositions.
A337665 counts compositions whose distinct parts are pairwise coprime.
A337666 ranks pairwise non-coprime compositions.
Cf. A007359, A007360, A087087, A302569, A302696, A304709, A305713, A335235, A337562, A337601, A337602, A337603.

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

A337983 Number of compositions of n into distinct parts, any two of which have a common divisor > 1.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 3, 1, 3, 3, 5, 1, 13, 1, 13, 7, 19, 1, 35, 1, 59, 15, 65, 1, 117, 5, 133, 27, 195, 1, 411, 7, 435, 67, 617, 17, 941, 7, 1177, 135, 1571, 13, 2939, 31, 3299, 375, 4757, 13, 6709, 43, 8813, 643, 11307, 61, 16427, 123, 24331, 1203, 30461, 67

Offset: 0

Gus Wiseman, Oct 06 2020

nonn

Number of pairwise non-coprime strict compositions of n.

			The a(2) = 1 through a(15) = 7 compositions (A..F = 10..15):
  2  3  4  5  6   7  8   9   A   B  C    D  E    F
              24     26  36  28     2A      2C   3C
              42     62  63  46     39      4A   5A
                             64     48      68   69
                             82     84      86   96
                                    93      A4   A5
                                    A2      C2   C3
                                    246     248
                                    264     284
                                    426     428
                                    462     482
                                    624     824
                                    642     842

A318717 is the unordered version.
A318719 is the version for Heinz numbers of partitions.
A337561 is the pairwise coprime instead of pairwise non-coprime version, or A337562 if singletons are considered coprime.
A337605*6 counts these compositions of length 3.
A337667 is the non-strict version, ranked by A337666.
A337696 ranks these compositions.
A051185 and A305843 (covering) count pairwise intersecting set-systems.
A101268 counts pairwise coprime or singleton compositions.
A200976 and A328673 are the unordered version.
A233564 ranks strict compositions.
A318749 is the version for factorizations, with non-strict version A319786.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A335236 ranks compositions neither a singleton nor pairwise coprime.
A337462 counts pairwise coprime compositions.
A337694 lists numbers with no two relatively prime prime indices.
Cf. A082024, A284825, A302797, A305713, A319752, A327516, A333227, A335235, A336737, A337604.

stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&&stabQ[#,CoprimeQ]&]],{n,0,30}]

A337987 Odd numbers whose distinct prime indices are pairwise coprime, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

15, 33, 35, 45, 51, 55, 69, 75, 77, 85, 93, 95, 99, 119, 123, 135, 141, 143, 145, 153, 155, 161, 165, 175, 177, 187, 201, 205, 207, 209, 215, 217, 219, 221, 225, 245, 249, 253, 255, 265, 275, 279, 287, 291, 295, 297, 309, 323, 327, 329, 335, 341, 355, 363, 369

Offset: 1

Gus Wiseman, Oct 23 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 integer partitions with no 1's whose distinct parts are pairwise coprime (A338315). The Heinz number of an integer 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:
     15: {2,3}      135: {2,2,2,3}    215: {3,14}
     33: {2,5}      141: {2,15}       217: {4,11}
     35: {3,4}      143: {5,6}        219: {2,21}
     45: {2,2,3}    145: {3,10}       221: {6,7}
     51: {2,7}      153: {2,2,7}      225: {2,2,3,3}
     55: {3,5}      155: {3,11}       245: {3,4,4}
     69: {2,9}      161: {4,9}        249: {2,23}
     75: {2,3,3}    165: {2,3,5}      253: {5,9}
     77: {4,5}      175: {3,3,4}      255: {2,3,7}
     85: {3,7}      177: {2,17}       265: {3,16}
     93: {2,11}     187: {5,7}        275: {3,3,5}
     95: {3,8}      201: {2,19}       279: {2,2,11}
     99: {2,2,5}    205: {3,13}       287: {4,13}
    119: {4,7}      207: {2,2,9}      291: {2,25}
    123: {2,13}     209: {5,8}        295: {3,17}

A304711 is the not necessarily odd version, with squarefree case A302797.
A337694 is a pairwise non-coprime instead of pairwise coprime version.
A337984 is the squarefree case.
A338315 counts the partitions with these Heinz numbers.
A338316 considers singletons coprime.
A007359 counts partitions into singleton or pairwise coprime parts with no 1's, with Heinz numbers A302568.
A304709 counts partitions whose distinct parts are pairwise coprime.
A327516 counts pairwise coprime partitions, with Heinz numbers A302696.
A337462 counts pairwise coprime compositions, ranked by A333227.
A337561 counts pairwise coprime strict compositions.
A337665 counts compositions whose distinct parts are pairwise coprime, ranked by A333228.
A337667 counts pairwise non-coprime compositions, ranked by A337666.
A337697 counts pairwise coprime compositions with no 1's.
A318717 counts pairwise non-coprime strict partitions, with Heinz numbers A318719.
Cf. A005408, A051424, A056239, A087087, A112798, A200976, A289508, A289509, A302569, A303282, A328867, A337485.

Select[Range[1,100,2],CoprimeQ@@Union[PrimePi/@First/@FactorInteger[#]]&]

A338315 Number of integer partitions of n with no 1's whose distinct parts are pairwise coprime, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 3, 2, 4, 4, 10, 6, 15, 13, 16, 21, 31, 29, 43, 41, 50, 63, 79, 81, 99, 113, 129, 145, 179, 197, 228, 249, 284, 328, 363, 418, 472, 522, 581, 655, 741, 828, 921, 1008, 1123, 1259, 1407, 1546, 1709, 1889, 2077, 2292, 2554, 2799, 3061, 3369

Offset: 0

Gus Wiseman, Oct 23 2020

nonn

The Heinz numbers of these partitions are given by A337987. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The a(5) = 1 through a(13) = 15 partitions (empty column indicated by dot, A = 10, B = 11):
  32   .  43    53    54     73     65      75      76
          52    332   72     433    74      543     85
          322         522    532    83      552     94
                      3222   3322   92      732     A3
                                    443     5322    B2
                                    533     33222   544
                                    722             553
                                    3332            733
                                    5222            922
                                    32222           4333
                                                    5332
                                                    7222
                                                    33322
                                                    52222
                                                    322222

A200976 is a pairwise non-coprime instead of pairwise coprime version.
A304709 allows 1's, with strict case A305713 and Heinz numbers A304711.
A318717 counts pairwise non-coprime strict partitions.
A337485 is the strict version, with Heinz numbers A337984.
A337987 gives the Heinz numbers of these partitions.
A338317 considers singletons coprime, with Heinz numbers A338316.
A007359 counts singleton or pairwise coprime partitions with no 1's.
A327516 counts pairwise coprime partitions, ranked by A302696.
A328673 counts partitions with no two distinct parts relatively prime.
A337462 counts pairwise coprime compositions, ranked by A333227.
A337561 counts pairwise coprime strict compositions.
A337665 counts compositions whose distinct parts are pairwise coprime.
A337667 counts pairwise non-coprime compositions, ranked by A337666.
A337697 counts pairwise coprime compositions with no 1's.
Cf. A051424, A101268, A220377, A302568, A328867, A333228, A337563.

Table[Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&CoprimeQ@@Union[#]&]],{n,0,30}]

A282748 Triangle read by rows: T(n,k) is the number of compositions of n into k parts x_1, x_2, ..., x_k such that gcd(x_i, x_j) = 1 for all i != j (where 1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 3, 4, 1, 1, 2, 9, 4, 5, 1, 1, 6, 3, 16, 5, 6, 1, 1, 4, 15, 4, 25, 6, 7, 1, 1, 6, 9, 28, 5, 36, 7, 8, 1, 1, 4, 21, 16, 45, 6, 49, 8, 9, 1, 1, 10, 9, 52, 25, 66, 7, 64, 9, 10, 1, 1, 4, 39, 16, 105, 36, 91, 8, 81, 10, 11, 1, 1, 12, 9, 100, 25, 186, 49, 120, 9, 100, 11, 12, 1, 1, 6, 45, 16, 205, 36, 301, 64, 153, 10, 121, 12, 13, 1

Offset: 1

N. J. A. Sloane, Mar 05 2017

See A101391 for the triangle T(n,k) = number of compositions of n into k parts x_1, x_2, ..., x_k such that gcd(x_1,x_2,...,x_k) = 1 (2 <= k <= n).

			Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  2,  3,   1;
  1,  4,  3,   4,   1;
  1,  2,  9,   4,   5,   1;
  1,  6,  3,  16,   5,   6,  1;
  1,  4, 15,   4,  25,   6,  7,   1;
  1,  6,  9,  28,   5,  36,  7,   8,  1;
  1,  4, 21,  16,  45,   6, 49,   8,  9,   1;
  1, 10,  9,  52,  25,  66,  7,  64,  9,  10,  1;
  1,  4, 39,  16, 105,  36, 91,   8, 81,  10, 11,  1;
  1, 12,  9, 100,  25, 186, 49, 120,  9, 100, 11, 12, 1;
  ...
From _Gus Wiseman_, Nov 12 2020: (Start)
Row n = 6 counts the following compositions:
  (6)  (15)  (114)  (1113)  (11112)  (111111)
       (51)  (123)  (1131)  (11121)
             (132)  (1311)  (11211)
             (141)  (3111)  (12111)
             (213)          (21111)
             (231)
             (312)
             (321)
             (411)
(End)

Alois P. Heinz, Rows n = 1..200, flattened (first 100 rows from Chai Wah Wu)
Temba Shonhiwa, Compositions with pairwise relatively prime summands within a restricted setting, Fibonacci Quart. 44 (2006), no. 4, 316-323.

A072704 counts the unimodal instead of coprime version.
A087087 and A335235 rank these compositions.
A101268 gives row sums.
A101391 is the relatively prime instead of pairwise coprime version.
A282749 is the unordered version.
A000740 counts relatively prime compositions, with strict case A332004.
A007360 counts pairwise coprime or singleton strict partitions.
A051424 counts pairwise coprime or singleton partitions, ranked by A302569.
A097805 counts compositions by sum and length.
A178472 counts compositions with a common divisor.
A216652 and A072574 count strict compositions by sum and length.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions, ranked by A302696.
A335235 ranks pairwise coprime or singleton compositions.
A337462 counts pairwise coprime compositions, ranked by A333227.
A337562 counts pairwise coprime or singleton strict compositions.
A337665 counts compositions whose distinct parts are pairwise coprime, ranked by A333228.
Cf. A000837, A007359, A302568, A337461, A337561, A337667.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],Length[#]==1||CoprimeQ@@#&]],{n,10},{k,n}] (* Gus Wiseman, Nov 12 2020 *)

It seems that no general formula or recurrence is known, although Shonhiwa gives formulas for a few of the early diagonals.

A337696 Numbers k such that the k-th composition in standard order (A066099) is strict and pairwise non-coprime, meaning the parts are distinct and any two of them have a common divisor > 1.

Original entry on oeis.org

0, 2, 4, 8, 16, 32, 34, 40, 64, 128, 130, 160, 256, 260, 288, 512, 514, 520, 544, 640, 1024, 2048, 2050, 2052, 2056, 2082, 2088, 2176, 2178, 2208, 2304, 2560, 2568, 2592, 4096, 8192, 8194, 8200, 8224, 8226, 8232, 8320, 8704, 8706, 8832, 10240, 10248, 10368

Offset: 1

Gus Wiseman, Oct 06 2020

nonn

Differs from A291165 in having 1090535424, corresponding to the composition (6,10,15).

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: ()        512: (10)       2304: (3,9)
       2: (2)       514: (8,2)      2560: (2,10)
       4: (3)       520: (6,4)      2568: (2,6,4)
       8: (4)       544: (4,6)      2592: (2,4,6)
      16: (5)       640: (2,8)      4096: (13)
      32: (6)      1024: (11)       8192: (14)
      34: (4,2)    2048: (12)       8194: (12,2)
      40: (2,4)    2050: (10,2)     8200: (10,4)
      64: (7)      2052: (9,3)      8224: (8,6)
     128: (8)      2056: (8,4)      8226: (8,4,2)
     130: (6,2)    2082: (6,4,2)    8232: (8,2,4)
     160: (2,6)    2088: (6,2,4)    8320: (6,8)
     256: (9)      2176: (4,8)      8704: (4,10)
     260: (6,3)    2178: (4,6,2)    8706: (4,8,2)
     288: (3,6)    2208: (4,2,6)    8832: (4,2,8)

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

A318719 gives the Heinz numbers of the unordered version, with non-strict version A337694.
A337667 counts the non-strict version.
A337983 counts these compositions, with unordered version A318717.
A051185 counts intersecting set-systems, with spanning case A305843.
A200976 and A328673 count the unordered non-strict version.
A337462 counts pairwise coprime compositions.
A318749 counts pairwise non-coprime factorizations, with strict case A319786.
All of the following pertain to compositions in standard order (A066099):
- A000120 is length.
- A070939 is sum.
- A124767 counts runs.
- A233564 ranks strict compositions.
- A272919 ranks constant compositions.
- A333219 is Heinz number.
- A333227 ranks pairwise coprime compositions, or A335235 if singletons are considered coprime.
- A333228 ranks compositions whose distinct parts are pairwise coprime.
- A335236 ranks compositions neither a singleton nor pairwise coprime.
- A337561 is the pairwise coprime instead of pairwise non-coprime version, or A337562 if singletons are considered coprime.
- A337666 ranks the non-strict version.
Cf. A082024, A101268, A302797, A305713, A319752, A327040, A327516, A336737, A337599, A337604, A337605.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
Select[Range[0,1000],UnsameQ@@stc[#]&&stabQ[stc[#],CoprimeQ]&]

Intersection of A337666 and A233564.

cp's OEIS Frontend

A337602 Number of ordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is always considered 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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A337984 Heinz numbers of pairwise coprime integer partitions with no 1's, where a singleton is not considered coprime.

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A337697 Number of pairwise coprime compositions of n with no 1's, where a singleton is not considered coprime.

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A337695 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are not pairwise coprime, where a singleton is always considered coprime.

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A337983 Number of compositions of n into distinct parts, any two of which have a common divisor > 1.

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A337987 Odd numbers whose distinct prime indices are pairwise coprime, where a singleton is not considered coprime unless it is (1).

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A338315 Number of integer partitions of n with no 1's whose distinct parts are pairwise coprime, where a singleton is not considered coprime unless it is (1).

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A282748 Triangle read by rows: T(n,k) is the number of compositions of n into k parts x_1, x_2, ..., x_k such that gcd(x_i, x_j) = 1 for all i != j (where 1 <= k <= n).

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