A332004 -id:A332004 - cp's OEIS frontend

A078374 Number of partitions of n into distinct and relatively prime parts.

1, 0, 1, 1, 2, 2, 4, 4, 6, 7, 11, 10, 17, 17, 23, 26, 37, 36, 53, 53, 70, 77, 103, 103, 139, 147, 184, 199, 255, 260, 339, 358, 435, 474, 578, 611, 759, 810, 963, 1045, 1259, 1331, 1609, 1726, 2015, 2200, 2589, 2762, 3259, 3509, 4058, 4416, 5119, 5488, 6364, 6882

Offset: 1

Vladeta Jovovic, Dec 24 2002

nonn

The Heinz numbers of these partitions are given by A302796, which is the intersection of A005117 (strict) and A289509 (relatively prime). - Gus Wiseman, Oct 18 2020

			From _Gus Wiseman_, Oct 18 2020: (Start)
The a(1) = 1 through a(13) = 17 partitions (empty column indicated by dot, A = 10, B = 11, C = 12):
  1   .  21   31   32   51    43    53    54    73     65     75     76
                   41   321   52    71    72    91     74     B1     85
                              61    431   81    532    83     543    94
                              421   521   432   541    92     651    A3
                                          531   631    A1     732    B2
                                          621   721    542    741    C1
                                                4321   632    831    643
                                                       641    921    652
                                                       731    5421   742
                                                       821    6321   751
                                                       5321          832
                                                                     841
                                                                     931
                                                                     A21
                                                                     5431
                                                                     6421
                                                                     7321
(End)

Seiichi Manyama, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms

Cf. A047966.
A000837 is the not necessarily strict version.
A302796 gives the Heinz numbers of these partitions.
A305713 is the pairwise coprime instead of relatively prime version.
A332004 is the ordered version.
A337452 is the case without 1's.
A000009 counts strict partitions.
A000740 counts relatively prime compositions.
Cf. A007359, A101268, A289508, A289509, A291166, A298748, A337451, A337485, A337451, A337561, A337563.

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

Moebius transform of A000009.

G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} (1 + x^n). - Ilya Gutkovskiy, Apr 26 2017

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

Original entry on oeis.org

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

A302698 Number of integer partitions of n into relatively prime parts that are all greater than 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 3, 2, 5, 4, 13, 7, 23, 18, 32, 33, 65, 50, 104, 92, 148, 153, 252, 226, 376, 376, 544, 570, 846, 821, 1237, 1276, 1736, 1869, 2552, 2643, 3659, 3887, 5067, 5509, 7244, 7672, 10086, 10909, 13756, 15168, 19195, 20735, 26237, 28708, 35418, 39207

Offset: 1

Gus Wiseman, Apr 11 2018

nonn

Two or more numbers are relatively prime if they have no common divisor other than 1. A single number is not considered relatively prime unless it is equal to 1 (which is impossible in this case).

The Heinz numbers of these partitions are given by A302697.

			The a(5) = 1 through a(12) = 7 partitions (empty column indicated by dot):
  (32)  .  (43)   (53)   (54)    (73)    (65)     (75)
           (52)   (332)  (72)    (433)   (74)     (543)
           (322)         (432)   (532)   (83)     (552)
                         (522)   (3322)  (92)     (732)
                         (3222)          (443)    (4332)
                                         (533)    (5322)
                                         (542)    (33222)
                                         (632)
                                         (722)
                                         (3332)
                                         (4322)
                                         (5222)
                                         (32222)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

A000837 is the version allowing 1's.
A002865 does not require relative primality.
A302697 gives the Heinz numbers of these partitions.
A337450 is the ordered version.
A337451 is the ordered strict version.
A337452 is the strict version.
A337485 is the pairwise coprime instead of relatively prime version.
A000740 counts relatively prime compositions.
A078374 counts relatively prime strict partitions.
A212804 counts compositions with no 1's.
A291166 appears to rank relatively prime compositions.
A332004 counts strict relatively prime compositions.
A337561 counts pairwise coprime strict compositions.
A338332 is the case of length 3, with strict case A338333.
Cf. A007359, A018783, A051424, A101268, A289508, A289509, A302568, A337563, A337984, A338468.

b:= proc(n, i, g) option remember; `if`(n=0, `if`(g=1, 1, 0),
      `if`(i<2, 0, b(n, i-1, g)+b(n-i, min(n-i, i), igcd(g, i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=1..60);  # Alois P. Heinz, Apr 12 2018

Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&GCD@@#===1&]],{n,30}]
(* Second program: *)
b[n_, i_, g_] := b[n, i, g] = If[n == 0, If[g == 1, 1, 0], If[i < 2, 0, b[n, i - 1, g] + b[n - i, Min[n - i, i], GCD[g, i]]]];
a[n_] := b[n, n, 0];
Array[a, 60] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

a(n) = A002865(n) - A018783(n).

Extended by Gus Wiseman, Oct 29 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.

A101271 Number of partitions of n into 3 distinct and relatively prime parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 8, 9, 12, 12, 16, 15, 21, 20, 26, 25, 33, 28, 40, 36, 45, 42, 56, 44, 65, 56, 70, 64, 84, 66, 96, 81, 100, 88, 120, 90, 133, 110, 132, 121, 161, 120, 175, 140, 176, 156, 208, 153, 220, 180, 222, 196, 261, 184, 280, 225, 270, 240, 312, 230, 341, 272

Offset: 6

Vladeta Jovovic, Dec 19 2004

The Heinz numbers of these partitions are the intersection of A289509 (relatively prime), A005117 (strict), and A014612 (triple). - Gus Wiseman, Oct 15 2020

			For n=10 we have 4 such partitions: 1+2+7, 1+3+6, 1+4+5 and 2+3+5.
From _Gus Wiseman_, Oct 13 2020: (Start)
The a(6) = 1 through a(18) = 15 triples (A..F = 10..15):
  321  421  431  432  532  542  543  643  653  654  754  764  765
            521  531  541  632  651  652  743  753  763  854  873
                 621  631  641  732  742  752  762  853  863  954
                      721  731  741  751  761  843  871  872  972
                           821  831  832  851  852  943  953  981
                                921  841  932  861  952  962  A53
                                     931  941  942  961  971  A71
                                     A21  A31  951  A51  A43  B43
                                          B21  A32  B32  A52  B52
                                               A41  B41  A61  B61
                                               B31  C31  B42  C51
                                               C21  D21  B51  D32
                                                         C32  D41
                                                         C41  E31
                                                         D31  F21
                                                         E21
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 6..10000

Cf. A023024-A023030, A000742-A000743, A023031-A023035.
A000741 is the ordered non-strict version.
A001399(n-6) does not require relative primality.
A023022 counts pairs instead of triples.
A023023 is the not necessarily strict version.
A078374 counts these partitions of any length, with Heinz numbers A302796.
A101271*6 is the ordered version.
A220377 is the pairwise coprime instead of relatively prime version.
A284825 counts the case that is pairwise non-coprime also.
A337605 is the pairwise non-coprime instead of relatively prime version.
A008289 counts strict partitions by sum and length.
A007304 gives the Heinz numbers of 3-part strict partitions.
A307719 counts 3-part pairwise coprime partitions.
A337601 counts 3-part partitions whose distinct parts are pairwise coprime.
Cf. A000010, A000217, A000837, A007360, A014612, A055684, A289509, A332004, A337452, A337563.

m:=3: with(numtheory): g:=sum(mobius(k)*x^(m*(m+1)/2*k)/Product(1-x^(i*k),i=1..m),k=1..20): gser:=series(g,x=0,80): seq(coeff(gser,x^n),n=6..77); # Emeric Deutsch, May 31 2005

Table[Length[Select[IntegerPartitions[n,{3}],UnsameQ@@#&&GCD@@#==1&]],{n,6,50}] (* Gus Wiseman, Oct 13 2020 *)

G.f. for the number of partitions of n into m distinct and relatively prime parts is Sum(moebius(k)*x^(m*(m+1)/2*k)/Product(1-x^(i*k), i=1..m), k=1..infinity).

More terms from Emeric Deutsch, May 31 2005

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}]

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.

A337452 Number of relatively prime strict integer partitions of n with no 1's.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 2, 1, 3, 2, 6, 3, 9, 7, 11, 11, 20, 15, 28, 24, 35, 36, 55, 47, 73, 71, 95, 96, 136, 123, 180, 177, 226, 235, 305, 299, 403, 406, 503, 523, 668, 662, 852, 873, 1052, 1115, 1370, 1391, 1720, 1784, 2125, 2252, 2701, 2786, 3348, 3520, 4116

Offset: 0

Gus Wiseman, Aug 31 2020

nonn

			The a(5) = 1 through a(16) = 11 partitions (A = 10, B = 11, C = 12, D = 13):
  32  43  53  54   73   65   75   76   95    87    97
      52      72   532  74   543  85   B3    B4    B5
              432       83   732  94   653   D2    D3
                        92        A3   743   654   754
                        542       B2   752   753   763
                        632       643  932   762   853
                                  652  5432  843   943
                                  742        852   952
                                  832        942   B32
                                             A32   6532
                                             6432  7432

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..300

A078374 is the version allowing 1's.
A302698 is the non-strict version.
A332004 is the ordered version allowing 1's.
A337450 is the ordered non-strict version.
A337451 is the ordered version.
A337485 is the pairwise coprime version.
A000837 counts relatively prime partitions.
A078374 counts relatively prime strict partitions.
A002865 counts partitions with no 1's.
A212804 counts compositions with no 1's.
A291166 appears to rank relatively prime compositions.
A337561 counts pairwise coprime strict compositions.
Cf. A007359, A101268, A289509, A337485, A337563.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,1]&&GCD@@#==1&]],{n,0,15}]

A337451 Number of relatively prime strict compositions of n with no 1's.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 4, 2, 10, 8, 20, 14, 34, 52, 72, 90, 146, 172, 244, 390, 502, 680, 956, 1218, 1686, 2104, 3436, 4078, 5786, 7200, 10108, 12626, 17346, 20876, 32836, 38686, 53674, 67144, 91528, 113426, 152810, 189124, 245884, 343350, 428494, 552548, 719156

Offset: 0

Gus Wiseman, Aug 31 2020

nonn

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

			The a(5) = 2 through a(10) = 8 compositions (empty column indicated by dot):
  (2,3)  .  (2,5)  (3,5)  (2,7)    (3,7)
  (3,2)     (3,4)  (5,3)  (4,5)    (7,3)
            (4,3)         (5,4)    (2,3,5)
            (5,2)         (7,2)    (2,5,3)
                          (2,3,4)  (3,2,5)
                          (2,4,3)  (3,5,2)
                          (3,2,4)  (5,2,3)
                          (3,4,2)  (5,3,2)
                          (4,2,3)
                          (4,3,2)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..350

A032022 does not require relative primality.
A302698 is the unordered non-strict version.
A332004 is the version allowing 1's.
A337450 is the non-strict version.
A337452 is the unordered version.
A000837 counts relatively prime partitions.
A032020 counts strict compositions.
A078374 counts strict relatively prime partitions.
A002865 counts partitions with no 1's.
A212804 counts compositions with no 1's.
A291166 appears to rank relatively prime compositions.
A337462 counts pairwise coprime compositions.
A337561 counts strict pairwise coprime compositions.
Cf. A000010, A007359, A101268, A178472, A216652, A289509, A337562, A337563.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,1]&&GCD@@#==1&]],{n,0,15}]

cp's OEIS Frontend

A078374 Number of partitions of n into distinct and relatively prime parts.

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

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A302698 Number of integer partitions of n into relatively prime parts that are all greater than 1.

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

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

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A101271 Number of partitions of n into 3 distinct and relatively prime parts.

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

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