A337462 -id:A337462 - cp's OEIS frontend

A101268 Number of compositions of n into pairwise relatively prime parts.

1, 1, 2, 4, 7, 13, 22, 38, 63, 101, 160, 254, 403, 635, 984, 1492, 2225, 3281, 4814, 7044, 10271, 14889, 21416, 30586, 43401, 61205, 85748, 119296, 164835, 226423, 309664, 422302, 574827, 781237, 1060182, 1436368, 1942589, 2622079, 3531152, 4742316, 6348411

Offset: 0

Vladeta Jovovic, Dec 18 2004

nonn

Here a singleton is always considered pairwise relatively prime. Compare to A337462. - Gus Wiseman, Oct 18 2020

			From _Gus Wiseman_, Oct 18 2020: (Start)
The a(1) = 1 through a(5) = 13 compositions:
  (1)  (2)   (3)    (4)     (5)
       (11)  (12)   (13)    (14)
             (21)   (31)    (23)
             (111)  (112)   (32)
                    (121)   (41)
                    (211)   (113)
                    (1111)  (131)
                            (311)
                            (1112)
                            (1121)
                            (1211)
                            (2111)
                            (11111)
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500 (terms 0..400 from Alois P. Heinz)
Temba Shonhiwa, Compositions with pairwise relatively prime summands within a restricted setting, Fibonacci Quart. 44 (2006), no. 4, 316-323.

Row sums of A282748.
A051424 is the unordered version, with strict case A007360.
A335235 ranks these compositions.
A337461 counts these compositions of length 3, with unordered version A307719 and unordered strict version A220377.
A337462 does not consider a singleton to be coprime unless it is (1), with strict version A337561.
A337562 is the strict case.
A337664 looks only at distinct parts, with non-constant version A337665.
A000740 counts relatively prime compositions, with strict case A332004.
A178472 counts compositions with a common factor.
Cf. A087087, A302569, A305713, A326675, A327516, A328673, A333227, A333228.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[#]<=1||CoprimeQ@@#&]],{n,0,10}] (* Gus Wiseman, Oct 18 2020 *)

It seems that no formula is known.

a(0)=1 prepended by Alois P. Heinz, Jun 14 2017

A337461 Number of pairwise coprime ordered triples of positive integers summing to n.

Original entry on oeis.org

0, 0, 0, 1, 3, 3, 9, 3, 15, 9, 21, 9, 39, 9, 45, 21, 45, 21, 87, 21, 93, 39, 87, 39, 153, 39, 135, 63, 153, 57, 255, 51, 207, 93, 225, 93, 321, 81, 291, 135, 321, 105, 471, 105, 393, 183, 381, 147, 597, 147, 531, 213, 507, 183, 759, 207, 621, 273, 621, 231

Offset: 0

Gus Wiseman, Sep 02 2020

nonn

			The a(3) = 1 through a(9) = 9 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)  (1,1,6)  (1,1,7)
           (1,2,1)  (1,3,1)  (1,2,3)  (1,5,1)  (1,2,5)  (1,3,5)
           (2,1,1)  (3,1,1)  (1,3,2)  (5,1,1)  (1,3,4)  (1,5,3)
                             (1,4,1)           (1,4,3)  (1,7,1)
                             (2,1,3)           (1,5,2)  (3,1,5)
                             (2,3,1)           (1,6,1)  (3,5,1)
                             (3,1,2)           (2,1,5)  (5,1,3)
                             (3,2,1)           (2,5,1)  (5,3,1)
                             (4,1,1)           (3,1,4)  (7,1,1)
                                               (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

A000212 counts the unimodal instead of coprime version.
A220377*6 is the strict case.
A307719 is the unordered version.
A337462 counts these compositions of any length.
A337563 counts the case of partitions with no 1's.
A337603 only requires the *distinct* parts to be pairwise coprime.
A337604 is the intersecting instead of coprime version.
A014612 ranks 3-part partitions.
A302696 ranks pairwise coprime partitions.
A327516 counts pairwise coprime partitions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A000217, A001399, A001840, A014311, A101268, A284825, A337562, A326675, A333227, A337601, A337602.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],CoprimeQ@@#&]],{n,0,30}]

A307719 Number of partitions of n into 3 mutually coprime parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 1, 3, 2, 4, 2, 7, 2, 8, 4, 8, 4, 15, 4, 16, 7, 15, 7, 26, 7, 23, 11, 26, 10, 43, 9, 35, 16, 38, 16, 54, 14, 49, 23, 54, 18, 79, 18, 66, 31, 64, 25, 100, 25, 89, 36, 85, 31, 127, 35, 104, 46, 104, 39, 167, 36, 125, 58, 129, 52, 185, 45

Offset: 0

Wesley Ivan Hurt, Apr 24 2019

The Heinz numbers of these partitions are the intersection of A014612 (triples) and A302696 (pairwise coprime). - Gus Wiseman, Oct 16 2020

			There are 2 partitions of 9 into 3 mutually coprime parts: 7+1+1 = 5+3+1, so a(9) = 2.
There are 4 partitions of 10 into 3 mutually coprime parts: 8+1+1 = 7+2+1 = 5+4+1 = 5+3+2, so a(10) = 4.
There are 2 partitions of 11 into 3 mutually coprime parts: 9+1+1 = 7+3+1, so a(11) = 2.
There are 7 partitions of 12 into 3 mutually coprime parts: 10+1+1 = 9+2+1 = 8+3+1 = 7+4+1 = 6+5+1 = 7+3+2 = 5+4+3, so a(12) = 7.

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Robert Israel)
Index entries for sequences related to partitions

A023022 is the version for pairs.
A220377 is the strict case, with ordered version A220377*6.
A327516 counts these partitions of any length, with strict version A305713 and Heinz numbers A302696.
A337461 is the ordered version.
A337563 is the case with no 1's.
A337599 is the pairwise non-coprime instead of pairwise coprime version.
A337601 only requires the distinct parts to be pairwise coprime.
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A002865 counts partitions with no 1's, with strict case A025147.
A007359 and A337485 count pairwise coprime partitions with no 1's.
A200976 and A328673 count pairwise non-coprime partitions.
Cf. A007304, A014612, A078374, A082024, A284825, A304709, A337462, A337605.

N:= 200: # to get a(0)..a(N)
A:= Array(0..N):
for a from 1 to N/3 do
  for b from a to (N-a)/2 do
    if igcd(a,b) > 1 then next fi;
    ab:= a*b;
    for c from b to N-a-b do
       if igcd(ab,c)=1 then A[a+b+c]:= A[a+b+c]+1 fi
od od od:
convert(A,list); # Robert Israel, May 09 2019

Table[Sum[Sum[Floor[1/(GCD[i, j] GCD[j, n - i - j] GCD[i, n - i - j])], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 100}]
Table[Length[Select[IntegerPartitions[n,{3}],CoprimeQ@@#&]],{n,0,100}] (* Gus Wiseman, Oct 15 2020 *)

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} [gcd(i,j) * gcd(j,n-i-j) * gcd(i,n-i-j) = 1], where [] is the Iverson bracket.

a(n > 2) = A220377(n) + 1. - Gus Wiseman, Oct 15 2020

A337561 Number of pairwise coprime strict compositions of n, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

1, 1, 0, 2, 2, 4, 8, 6, 16, 12, 22, 40, 40, 66, 48, 74, 74, 154, 210, 228, 242, 240, 286, 394, 806, 536, 840, 654, 1146, 1618, 2036, 2550, 2212, 2006, 2662, 4578, 4170, 7122, 4842, 6012, 6214, 11638, 13560, 16488, 14738, 15444, 16528, 25006, 41002, 32802

Offset: 0

Gus Wiseman, Sep 18 2020

nonn

			The a(1) = 1 through a(9) = 12 compositions (empty column shown as dot):
   (1)  .  (1,2)  (1,3)  (1,4)  (1,5)    (1,6)  (1,7)    (1,8)
           (2,1)  (3,1)  (2,3)  (5,1)    (2,5)  (3,5)    (2,7)
                         (3,2)  (1,2,3)  (3,4)  (5,3)    (4,5)
                         (4,1)  (1,3,2)  (4,3)  (7,1)    (5,4)
                                (2,1,3)  (5,2)  (1,2,5)  (7,2)
                                (2,3,1)  (6,1)  (1,3,4)  (8,1)
                                (3,1,2)         (1,4,3)  (1,3,5)
                                (3,2,1)         (1,5,2)  (1,5,3)
                                                (2,1,5)  (3,1,5)
                                                (2,5,1)  (3,5,1)
                                                (3,1,4)  (5,1,3)
                                                (3,4,1)  (5,3,1)
                                                (4,1,3)
                                                (4,3,1)
                                                (5,1,2)
                                                (5,2,1)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..600

A072706 counts unimodal strict compositions.
A220377*6 counts these compositions of length 3.
A305713 is the unordered version.
A337462 is the not necessarily strict version.
A000740 counts relatively prime compositions, with strict case A332004.
A051424 counts pairwise coprime or singleton partitions.
A101268 considers all singletons to be coprime, with strict case A337562.
A178472 counts compositions with a common factor > 1.
A327516 counts pairwise coprime partitions, with strict case A305713.
A328673 counts pairwise non-coprime partitions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A007359, A007360, A087087, A216652, A220377, A302569, A307719, A326675, A333227, A337461.

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

a(n) = A337562(n) - 1 for n > 1.

A337485 Number of pairwise coprime integer partitions of n with no 1's, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 2, 1, 2, 2, 4, 3, 5, 4, 4, 7, 8, 9, 10, 10, 9, 13, 17, 18, 17, 19, 19, 24, 29, 34, 33, 31, 31, 42, 42, 56, 55, 50, 54, 66, 77, 86, 86, 79, 81, 96, 124, 127, 126, 127, 126, 145, 181, 190, 184, 183, 192, 212, 262, 289, 278, 257, 270, 311

Offset: 0

Gus Wiseman, Sep 21 2020

nonn

Such a partition is necessarily strict.

The Heinz numbers of these partitions are the intersection of A005408 (no 1's), A005117 (strict), and A302696 (coprime).

			The a(n) partitions for n = 5, 7, 12, 13, 16, 17, 18, 19 (A..H = 10..17):
  (3,2)  (4,3)  (7,5)    (7,6)  (9,7)    (9,8)      (B,7)    (A,9)
         (5,2)  (5,4,3)  (8,5)  (B,5)    (A,7)      (D,5)    (B,8)
                (7,3,2)  (9,4)  (D,3)    (B,6)      (7,6,5)  (C,7)
                         (A,3)  (7,5,4)  (C,5)      (8,7,3)  (D,6)
                         (B,2)  (8,5,3)  (D,4)      (9,5,4)  (E,5)
                                (9,5,2)  (E,3)      (9,7,2)  (F,4)
                                (B,3,2)  (F,2)      (B,4,3)  (G,3)
                                         (7,5,3,2)  (B,5,2)  (H,2)
                                                    (D,3,2)  (B,5,3)
                                                             (7,5,4,3)

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

A005408 intersected with A302696 ranks these partitions.
A007359 considers all singletons to be coprime.
A327516 allows 1's, with non-strict version A305713.
A337452 is the relatively prime instead of pairwise coprime version, with non-strict version A302698.
A337563 is the restriction to partitions of length 3.
A002865 counts partitions with no 1's.
A078374 counts relatively prime strict partitions.
A200976 and A328673 count pairwise non-coprime partitions.
Cf. A101268, A220377, A302696, A304709, A332004, A337450, A337451, A337462, A337561, A337605.

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

a(n) = A007359(n) - 1 for n > 1.

A055684 Number of different n-pointed stars.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 1, 4, 1, 5, 2, 3, 3, 7, 2, 8, 3, 5, 4, 10, 3, 9, 5, 8, 5, 13, 3, 14, 7, 9, 7, 11, 5, 17, 8, 11, 7, 19, 5, 20, 9, 11, 10, 22, 7, 20, 9, 15, 11, 25, 8, 19, 11, 17, 13, 28, 7, 29, 14, 17, 15, 23, 9, 32, 15, 21, 11, 34, 11, 35, 17, 19, 17, 29, 11

Offset: 3

Robert G. Wilson v, Jun 09 2000

Does not count rotations or reflections.
This is also the distinct ways of writing a number as the sum of two positive integers greater than one that are coprimes. - Lei Zhou, Mar 19 2014
Equivalently, a(n) is the number of relatively prime 2-part partitions of n without 1's. The Heinz numbers of these partitions are the intersection of A001358 (pairs), A005408 (no 1's), and A000837 (relatively prime) or A302696 (pairwise coprime). - Gus Wiseman, Oct 28 2020

			The first star has five points and is unique. The next is the seven pointed star and it comes in two varieties.
From _Gus Wiseman_, Oct 28 2020: (Start)
The a(5) = 1 through a(17) = 7 irreducible pairs > 1 (shown as fractions, empty column indicated by dot):
  2/3  .  2/5  3/5  2/7  3/7  2/9  5/7  2/11  3/11  2/13  3/13  2/15
          3/4       4/5       3/8       3/10  5/9   4/11  5/11  3/14
                              4/7       4/9         7/8   7/9   4/13
                              5/6       5/8                     5/12
                                        6/7                     6/11
                                                                7/10
                                                                8/9
(End)

Mark A. Herkommer, "Number Theory, A Programmer's Guide," McGraw-Hill, New York, 1999, page 58.

Lei Zhou, Table of n, a(n) for n = 3..10002
Alexander Bogomolny, Polygons: formality and intuition.. Includes applet to draw star polygons.
Vi Hart, Doodling in Math Class: Stars, Video (2010).
Hugo Pfoertner, Star-shaped regular polygons up to n=25.
Eric Weisstein's World of Mathematics, Star Polygon

Cf. A023022.
Cf. A053669 smallest skip increment, A102302 skip increment of densest star polygon.
A055684*2 is the ordered version.
A082023 counts the complement (reducible pairs > 1).
A220377, A337563, and A338332 count triples instead of pairs.
A000837 counts relatively prime partitions, with strict case A078374.
A002865 counts partitions with no 1's, with strict case A025147.
A007359 and A337485 count pairwise coprime partitions with no 1's.
A302698 counts relatively prime partitions with no 1's, with strict case A337452.
A327516 counts pairwise coprime partitions, with strict case A305713.
A337450 counts relatively prime compositions with no 1's, with strict case A337451.
Cf. A001399, A101268, A140106, A337461, A337462, A338333.

with(numtheory): A055684 := n->(phi(n)-2)/2; seq(A055684(n), n=3..100);

Table[(EulerPhi[n]-2)/2, {n, 3, 50}]
Table[Length[Select[IntegerPartitions[n,{2}],!MemberQ[#,1]&&CoprimeQ@@#&]],{n,0,30}] (* Gus Wiseman, Oct 28 2020 *)

a(n) = A023022(n) - 1.

a(n) + A082023(n) = A140106(n). - Gus Wiseman, Oct 28 2020

A337563 Number of pairwise coprime unordered triples of positive integers > 1 summing to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 4, 0, 7, 1, 7, 3, 9, 2, 15, 3, 13, 5, 17, 4, 29, 5, 20, 8, 28, 8, 42, 8, 31, 14, 42, 10, 59, 12, 45, 21, 52, 14, 77, 17, 68, 26, 69, 19, 101, 26, 84, 34, 86, 25, 138, 28, 95, 43, 111, 36, 161, 35, 118, 52, 151

Offset: 0

Gus Wiseman, Sep 21 2020

nonn

Such partitions are necessarily strict.

The Heinz numbers of these partitions are the intersection of A005408 (no 1's), A014612 (triples), and A302696 (coprime).

			The a(10) = 1 through a(24) = 15 triples (empty columns indicated by dots, A..J = 10..19):
  532  .  543  .  743  753  754  .  765  B53  875  975  985  B75  987
          732     752       853     873       974  B73  B65  D73  B76
                            952     954       A73  D53  B74       B85
                            B32     972       B54       B83       B94
                                    B43       B72       B92       BA3
                                    B52       D43       D54       C75
                                    D32       D52       D72       D65
                                                        E53       D74
                                                        H32       D83
                                                                  D92
                                                                  F72
                                                                  G53
                                                                  H43
                                                                  H52
                                                                  J32

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

A055684 is the version for pairs.
A220377 allows 1's, with non-strict version A307719.
A337485 counts these partitions of any length.
A337563*6 is the ordered version.
A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.
A002865 counts partitions with no 1's, with strict case A025147.
A007359 counts pairwise coprime partitions with no 1's.
A078374 counts relatively prime strict partitions.
A200976 and A328673 count pairwise non-coprime partitions.
A302696 ranks pairwise coprime partitions.
A302698 counts relatively prime partitions with no 1's.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions.
A337452 counts relatively prime strict partitions with no 1's.
Cf. A007304, A082024, A101268, A284825, A332004, A337451, A337461, A337462, A337561, A337599, A337601, A337605.

Table[Length[Select[IntegerPartitions[n,{3}],!MemberQ[#,1]&&CoprimeQ@@#&]],{n,0,30}]

A337667 Number of compositions of n where any two parts have a common divisor > 1.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 5, 1, 8, 4, 17, 1, 38, 1, 65, 19, 128, 1, 284, 1, 518, 67, 1025, 1, 2168, 16, 4097, 256, 8198, 1, 16907, 7, 32768, 1027, 65537, 79, 133088, 19, 262145, 4099, 524408, 25, 1056731, 51, 2097158, 16636, 4194317, 79, 8421248, 196, 16777712

Offset: 0

Gus Wiseman, Oct 05 2020

nonn

First differs from A178472 at a(31) = 7, a(31) = 1.

			The a(2) = 1 through a(10) = 17 compositions (A = 10):
   2   3   4    5   6     7   8      9     A
           22       24        26     36    28
                    33        44     63    46
                    42        62     333   55
                    222       224          64
                              242          82
                              422          226
                              2222         244
                                           262
                                           424
                                           442
                                           622
                                           2224
                                           2242
                                           2422
                                           4222
                                           22222

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..290

A101268 = 1 + A337462 is the pairwise coprime version.
A328673 = A200976 + 1 is the unordered version.
A337604 counts these compositions of length 3.
A337666 ranks these compositions.
A337694 gives Heinz numbers of the unordered version.
A337983 is the strict case.
A051185 counts intersecting set-systems, with spanning case A305843.
A318717 is the unordered strict case.
A319786 is the version for factorizations, with strict case A318749.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A051424, A082024, A178472, A284825, A319752, A327039, A337599, A337605, A337665.

stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],stabQ[#,CoprimeQ]&]],{n,0,15}]

A332004 Number of compositions (ordered partitions) of n into distinct and relatively prime parts.

Original entry on oeis.org

1, 1, 0, 2, 2, 4, 8, 12, 16, 24, 52, 64, 88, 132, 180, 344, 416, 616, 816, 1176, 1496, 2736, 3232, 4756, 6176, 8756, 11172, 15576, 24120, 30460, 41456, 55740, 74440, 97976, 130192, 168408, 256464, 315972, 429888, 558192, 749920, 958264, 1274928, 1621272, 2120288, 3020256

Offset: 0

Ilya Gutkovskiy, Feb 04 2020

nonn

Moebius transform of A032020.

Ranking these compositions using standard compositions (A066099) gives the intersection of A233564 (strict) with A291166 (relatively prime). - Gus Wiseman, Oct 18 2020

			a(6) = 8 because we have [5, 1], [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 1, 3], [1, 5], [1, 3, 2] and [1, 2, 3].
From _Gus Wiseman_, Oct 18 2020: (Start)
The a(1) = 1 through a(8) = 16 compositions (empty column indicated by dot):
  (1)  .  (1,2)  (1,3)  (1,4)  (1,5)    (1,6)    (1,7)
          (2,1)  (3,1)  (2,3)  (5,1)    (2,5)    (3,5)
                        (3,2)  (1,2,3)  (3,4)    (5,3)
                        (4,1)  (1,3,2)  (4,3)    (7,1)
                               (2,1,3)  (5,2)    (1,2,5)
                               (2,3,1)  (6,1)    (1,3,4)
                               (3,1,2)  (1,2,4)  (1,4,3)
                               (3,2,1)  (1,4,2)  (1,5,2)
                                        (2,1,4)  (2,1,5)
                                        (2,4,1)  (2,5,1)
                                        (4,1,2)  (3,1,4)
                                        (4,2,1)  (3,4,1)
                                                 (4,1,3)
                                                 (4,3,1)
                                                 (5,1,2)
                                                 (5,2,1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to compositions

Cf. A007360, A032020, A108700, A302698.
A000740 is the non-strict version.
A078374 is the unordered version (non-strict: A000837).
A101271*6 counts these compositions of length 3 (non-strict: A000741).
A337561/A337562 is the pairwise coprime instead of relatively prime version (non-strict: A337462/A101268).
A289509 gives the Heinz numbers of relatively prime partitions.
A333227/A335235 ranks pairwise coprime compositions.
Cf. A001523, A178472, A216652, A289508, A291166, A333228.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&GCD@@#<=1&]],{n,0,15}] (* Gus Wiseman, Oct 18 2020 *)

A337562 Number of pairwise coprime strict compositions of n, where a singleton is always considered coprime.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 9, 7, 17, 13, 23, 41, 41, 67, 49, 75, 75, 155, 211, 229, 243, 241, 287, 395, 807, 537, 841, 655, 1147, 1619, 2037, 2551, 2213, 2007, 2663, 4579, 4171, 7123, 4843, 6013, 6215, 11639, 13561, 16489, 14739, 15445, 16529, 25007, 41003, 32803

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

			The a(1) = 1 through a(9) = 12 compositions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)    (8)      (9)
            (1,2)  (1,3)  (1,4)  (1,5)    (1,6)  (1,7)    (1,8)
            (2,1)  (3,1)  (2,3)  (5,1)    (2,5)  (3,5)    (2,7)
                          (3,2)  (1,2,3)  (3,4)  (5,3)    (4,5)
                          (4,1)  (1,3,2)  (4,3)  (7,1)    (5,4)
                                 (2,1,3)  (5,2)  (1,2,5)  (7,2)
                                 (2,3,1)  (6,1)  (1,3,4)  (8,1)
                                 (3,1,2)         (1,4,3)  (1,3,5)
                                 (3,2,1)         (1,5,2)  (1,5,3)
                                                 (2,1,5)  (3,1,5)
                                                 (2,5,1)  (3,5,1)
                                                 (3,1,4)  (5,1,3)
                                                 (3,4,1)  (5,3,1)
                                                 (4,1,3)
                                                 (4,3,1)
                                                 (5,1,2)
                                                 (5,2,1)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..600

A007360 is the unordered version, with non-strict version A051424.
A101268 is the not necessarily strict version.
A220377*6 counts these compositions of length 3.
A337561 does not consider a singleton to be coprime unless it is (1), with non-strict version A337462.
A337664 looks only at distinct parts.
A000740 counts relatively prime compositions, with strict case A332004.
A072706 counts unimodal strict compositions.
A178472 counts compositions with a common factor.
A327516 counts pairwise coprime partitions, with strict case A305713.
A328673 counts pairwise non-coprime partitions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A087087, A220377, A302569, A307719, A326675, A333227, A335235, A335238, A337461, A337665.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&(Length[#]<=1||CoprimeQ@@#)&]],{n,0,10}]

a(n > 1) = A337561(n) + 1 for n > 1.

cp's OEIS Frontend

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