A337601 -id:A337601 - cp's OEIS frontend

A220377 Number of partitions of n into three distinct and mutually relatively prime parts.

1, 0, 2, 1, 3, 1, 6, 1, 7, 3, 7, 3, 14, 3, 15, 6, 14, 6, 25, 6, 22, 10, 25, 9, 42, 8, 34, 15, 37, 15, 53, 13, 48, 22, 53, 17, 78, 17, 65, 30, 63, 24, 99, 24, 88, 35, 84, 30, 126, 34, 103, 45, 103, 38, 166, 35, 124, 57, 128, 51, 184, 44, 150, 67, 172, 52, 218

Offset: 6

Carl Najafi, Dec 13 2012

nonn

The Heinz numbers of these partitions are the intersection of A005117 (strict), A014612 (triples), and A302696 (coprime). - Gus Wiseman, Oct 14 2020

			For n=10 we have three such partitions: 1+2+7, 1+4+5 and 2+3+5.
From _Gus Wiseman_, Oct 14 2020: (Start)
The a(6) = 1 through a(20) = 15 triples (empty column indicated by dot, A..H = 10..17):
321  .  431  531  532  731  543  751  743  753  754  971  765  B53  875
        521       541       651       752  951  853  B51  873  B71  974
                  721       732       761  B31  871  D31  954  D51  A73
                            741       851       952       972       A91
                            831       941       B32       981       B54
                            921       A31       B41       A71       B72
                                      B21       D21       B43       B81
                                                          B52       C71
                                                          B61       D43
                                                          C51       D52
                                                          D32       D61
                                                          D41       E51
                                                          E31       F41
                                                          F21       G31
                                                                    H21
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 6..10000 (terms 6..1000 from Seiichi Manyama)

Cf. A015617, A300815.
A023022 is the 2-part version.
A101271 is the relative prime instead of pairwise coprime version.
A220377*6 is the ordered version.
A305713 counts these partitions of any length, with Heinz numbers A302797.
A307719 is the non-strict version.
A337461 is the non-strict ordered version.
A337563 is the case with no 1's.
A337605 is the pairwise non-coprime instead of pairwise coprime version.
A001399(n-6) counts strict 3-part partitions, with Heinz numbers A007304.
A008284 counts partitions by sum and length, with strict case A008289.
A318717 counts pairwise non-coprime strict partitions.
A326675 ranks pairwise coprime sets.
A327516 counts pairwise coprime partitions.
A337601 counts 3-part partitions whose distinct parts are pairwise coprime.
Cf. A000217, A007360, A023023, A051424, A078374, A087087, A302696, A333227, A337485, A337561.

Table[Length@Select[ IntegerPartitions[ n, {3}], #[[1]] != #[[2]] != #[[3]] && GCD[#[[1]], #[[2]]] == 1 && GCD[#[[1]], #[[3]]] == 1 && GCD[#[[2]], #[[3]]] == 1 &], {n, 6, 100}]
Table[Count[IntegerPartitions[n,{3}],?(CoprimeQ@@#&&Length[ Union[#]] == 3&)],{n,6,100}] (* _Harvey P. Dale, May 22 2020 *)

a(n)=my(P=partitions(n));sum(i=1,#P,#P[i]==3&&P[i][1]Charles R Greathouse IV, Dec 14 2012

a(n > 2) = A307719(n) - 1. - Gus Wiseman, Oct 15 2020

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

A023023 Number of partitions of n into 3 unordered relatively prime parts.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 6, 10, 8, 14, 12, 16, 16, 24, 18, 30, 24, 32, 30, 44, 32, 50, 42, 54, 48, 70, 48, 80, 64, 80, 72, 96, 72, 114, 90, 112, 96, 140, 96, 154, 120, 144, 132, 184, 128, 196, 150, 192, 168, 234, 162, 240, 192, 240, 210, 290, 192, 310, 240, 288, 256, 336, 240, 374

Offset: 3

David W. Wilson

nonn

			From _Gus Wiseman_, Oct 08 2020: (Start)
The a(3) = 1 through a(13) = 14 triples (A = 10, B = 11):
  111   211   221   321   322   332   432   433   443   543   544
              311   411   331   431   441   532   533   552   553
                          421   521   522   541   542   651   643
                          511   611   531   631   551   732   652
                                      621   721   632   741   661
                                      711   811   641   831   733
                                                  722   921   742
                                                  731   A11   751
                                                  821         832
                                                  911         841
                                                              922
                                                              931
                                                              A21
                                                              B11
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 3..10000
Mohamed El Bachraoui, Relatively Prime Partitions with Two and Three Parts, Fibonacci Quart. 46/47 (2008/2009), no. 4, 341-345.

Cf. A023024-A023030, A000742-A000743, A023032-A023035.
A000741 is the ordered version.
A000837 counts these partitions of any length.
A001399(n-3) does not require relative primality.
A023022 is the 2-part version.
A101271 is the strict case.
A284825 counts the case that is also pairwise non-coprime.
A289509 intersected with A014612 gives the Heinz numbers.
A307719 is the pairwise coprime instead of relatively prime version.
A337599 is the pairwise non-coprime instead of relative prime version.
A008284 counts partitions by sum and length.
A078374 counts relatively prime strict partitions.
A337601 counts 3-part partitions whose distinct parts are pairwise coprime.
Cf. A000010, A000217, A007434, A055684, A078374, A200976, A220377, A302698, A327516, A337563, A337600, A337605.

Table[Length[Select[IntegerPartitions[n,{3}],GCD@@#==1&]],{n,3,50}] (* Gus Wiseman, Oct 08 2020 *)

G.f. for the number of partitions of n into m unordered relatively prime parts is Sum(moebius(k)*x^(m*k)/Product(1-x^(i*k), i=1..m), k=1..infinity). - Vladeta Jovovic, Dec 21 2004
a(n) = (n^2/12)*Product_{prime p|n} (1 - 1/p^2) = A007434(n)/12 for n > 3 (proved by Mohamed El Bachraoui). [Jonathan Sondow, May 27 2009]
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} floor(1/gcd(i,k,n-i-k)). - Wesley Ivan Hurt, Jan 02 2021

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}]

A337603 Number of ordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 9, 9, 18, 15, 24, 21, 42, 24, 51, 30, 54, 42, 93, 45, 102, 54, 99, 69, 162, 66, 150, 87, 168, 96, 264, 93, 228, 120, 246, 126, 336, 132, 315, 168, 342, 162, 486, 165, 420, 216, 411, 213, 618, 207, 558, 258, 540, 258, 783, 264, 654, 324, 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)  (2,3,1)  (3,1,3)  (1,6,1)
                             (3,1,2)  (3,2,2)  (2,1,5)
                             (3,2,1)  (3,3,1)  (2,3,3)
                             (4,1,1)  (5,1,1)  (2,5,1)
                                               (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

A014311 intersected with A333228 ranks these compositions.
A220377*6 is the strict case.
A337461 is the strict case except for any number of 1's.
A337601 is the unordered version.
A337602 considers all singletons to be coprime.
A337665 counts these compositions of any length, ranked by A333228 with complement A335238.
A000217(n - 2) counts 3-part compositions.
A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.
A007318 and A097805 count compositions by length.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A304711 ranks partitions whose distinct parts are pairwise coprime.
A305713 counts strict pairwise coprime partitions.
A327516 counts pairwise coprime partitions, with strict case A305713.
A333227 ranks pairwise coprime compositions.
Cf. A000740, A001840, A007359, A087087, A178472, A284825, A302696, A307719, A335235, A337561, A337695.

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

A337665 Number of compositions of n whose distinct parts are pairwise coprime, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

0, 1, 1, 3, 6, 15, 27, 57, 108, 208, 393, 749, 1415, 2687, 5076, 9583, 18088, 34156, 64511, 121898, 230368, 435460, 823376, 1557420, 2946931, 5578109, 10561987, 20005126, 37902509, 71832372, 136173266, 258211602, 489738622, 929074445, 1762899107, 3345713031

Offset: 0

Gus Wiseman, Sep 22 2020

nonn

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

			The a(1) = 1 through a(5) = 15 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)  (1,2,2)
                                  (1,3,1)
                                  (2,1,2)
                                  (2,2,1)
                                  (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..220

A000740 is a relatively prime instead of pairwise coprime version.
A304709 is the unordered version.
A333228 ranks these compositions.
A337561 is the strict case.
A337603 is the length-3 case.
A337664 considers all singletons to be coprime.
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.
A337461 counts pairwise coprime length-3 compositions.
Cf. A007359, A007360, A302569, A302696, A304712, A335235, A335238, A337562, A337600, A337601, A337602, A337695.

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

a(26)-a(35) from Alois P. Heinz, Sep 29 2020

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.

Original entry on oeis.org

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}]

A100565 a(n) = Card{(x,y,z) : x <= y <= z, x|n, y|n, z|n, gcd(x,y)=1, gcd(x,z)=1, gcd(y,z)=1}.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 4, 3, 5, 2, 8, 2, 5, 5, 5, 2, 8, 2, 8, 5, 5, 2, 11, 3, 5, 4, 8, 2, 15, 2, 6, 5, 5, 5, 13, 2, 5, 5, 11, 2, 15, 2, 8, 8, 5, 2, 14, 3, 8, 5, 8, 2, 11, 5, 11, 5, 5, 2, 25, 2, 5, 8, 7, 5, 15, 2, 8, 5, 15, 2, 18, 2, 5, 8, 8, 5, 15, 2, 14, 5, 5, 2, 25, 5, 5, 5, 11, 2, 25, 5, 8, 5, 5, 5, 17

Offset: 1

Vladeta Jovovic, Nov 28 2004

First differs from A018892 at a(30) = 15, A018892(30) = 14.
First differs from A343654 at a(210) = 51, A343654(210) = 52.
Also a(n) = Card{(x,y,z) : x <= y <= z and lcm(x,y)=n, lcm(x,z)=n, lcm(y,z)=n}.
In words, a(n) is the number of pairwise coprime unordered triples of divisors of n. - Gus Wiseman, May 01 2021

			From _Gus Wiseman_, May 01 2021: (Start)
The a(n) triples for n = 1, 2, 4, 6, 8, 12, 24:
  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)   (1,1,1)
           (1,1,2)  (1,1,2)  (1,1,2)  (1,1,2)  (1,1,2)   (1,1,2)
                    (1,1,4)  (1,1,3)  (1,1,4)  (1,1,3)   (1,1,3)
                             (1,1,6)  (1,1,8)  (1,1,4)   (1,1,4)
                             (1,2,3)           (1,1,6)   (1,1,6)
                                               (1,2,3)   (1,1,8)
                                               (1,3,4)   (1,2,3)
                                               (1,1,12)  (1,3,4)
                                                         (1,3,8)
                                                         (1,1,12)
                                                         (1,1,24)
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A070919, A086222, A086165, A048691, A063647.
Positions of 2's through 5's are A000040, A001248, A030078, A068993.
The version for subsets of {1..n} instead of divisors is A015617.
The version for pairs of divisors is A018892.
The ordered version is A048785.
The strict case is A066620.
The version for strict partitions is A220377.
A version for sets of divisors of any size is A225520.
The version for partitions is A307719 (no 1's: A337563).
The case of distinct parts coprime is A337600 (ordered: A337602).
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A007304 ranks 3-part strict partitions.
A014311 ranks 3-part compositions.
A014612 ranks 3-part partitions.
A051026 counts pairwise indivisible subsets of {1..n}.
A302696 lists Heinz numbers of pairwise coprime partitions.
A337461 counts 3-part pairwise coprime compositions.
Cf. A000961, A000977, A007360, A023022, A087087, A276187, A282935, A337601, A337603, A338331, A343652, A343654.

pwcop[y_]:=And@@(GCD@@#==1&/@Subsets[y,{2}]);
Table[Length[Select[Tuples[Divisors[n],3],LessEqual@@#&&pwcop[#]&]],{n,30}] (* Gus Wiseman, May 01 2021 *)

A100565(n) = (numdiv(n^3)+3*numdiv(n)+2)/6; \\ Antti Karttunen, May 19 2017

a(n) = (tau(n^3) + 3*tau(n) + 2)/6.

cp's OEIS Frontend

A220377 Number of partitions of n into three distinct and mutually relatively prime parts.

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

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

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A023023 Number of partitions of n into 3 unordered relatively prime parts.

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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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A337603 Number of ordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is not considered coprime unless it is (1).

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A337665 Number of compositions of n whose distinct parts are pairwise coprime, where a singleton is not considered coprime unless it is (1).

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