A023023 -id:A023023 - 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

A211540 Number of ordered triples (w,x,y) with all terms in {1..n} and 2w = 3x + 4y.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 56, 61, 65, 70, 75, 80, 85, 91, 96, 102, 108, 114, 120, 127, 133, 140, 147, 154, 161, 169, 176, 184, 192, 200, 208, 217, 225, 234, 243, 252, 261, 271, 280, 290

Offset: 0

Clark Kimberling, Apr 15 2012

For a guide to related sequences, see A211422.
Also the number of partitions of n+1 into three parts, where each part > 1. - Peter Woodward, May 25 2015
a(n) is also equal to the number of partitions of n+4 into three distinct parts, where each part > 1. - Giovanni Resta, May 26 2015
Number of different distributions of n+1 identical balls in 3 boxes as x,y,z where 0 < x < y < z. - Ece Uslu and Esin Becenen, Dec 31 2015
After the first three terms, partial sums of A008615. - Robert Israel, Dec 31 2015
For n >= 2, also the number of partitions of n - 2 into 3 parts. The Heinz numbers of these partitions are given by A014612. - Gus Wiseman, Oct 11 2020

			a(5) = a(6) = 1 with only one ordered triple (5,2,1). - _Michael Somos_, Feb 02 2015
a(11) = 5 Number of different distributions of 11 identical balls in 3 boxes as x,y and z where 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Dec 31 2015
a(1) = a(2) = a(3) = a(4) = a(5) = 0, since with fewer than 6 identical balls there is no such distribution with 3 boxes that holds for 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Dec 31 2015
G.f.: x^5 + x^6 + 2*x^7 + 3*x^8 + 4*x^9 + 5*x^10 + 7*x^11 + 8*x^12 + ...
From _Gus Wiseman_, Oct 11 2020: (Start)
The a(5) = 1 through a(15) = 14 partitions of n + 1 into three parts > 1 [Woodward] are the following (A = 10, B = 11, C = 12). The ordered version is A000217(n - 4) and the Heinz numbers are A046316.
  222  322  332  333  433  443  444  544  554  555  655
            422  432  442  533  543  553  644  654  664
                 522  532  542  552  643  653  663  754
                      622  632  633  652  662  744  763
                           722  642  733  743  753  772
                                732  742  752  762  844
                                822  832  833  843  853
                                     922  842  852  862
                                          932  933  943
                                          A22  942  952
                                               A32  A33
                                               B22  A42
                                                    B32
                                                    C22
The a(5) = 1 through a(15) = 14 partitions of n + 4 into three distinct parts > 1 [Resta] are the following (A = 10, B = 11, C = 12, D = 13, E = 14). The ordered version is A211540*6 and the Heinz numbers are A046389.
  432  532  542  543  643  653  654  754  764  765  865
            632  642  652  743  753  763  854  864  874
                 732  742  752  762  853  863  873  964
                      832  842  843  862  872  954  973
                           932  852  943  953  963  982
                                942  952  962  972  A54
                                A32  A42  A43  A53  A63
                                     B32  A52  A62  A72
                                          B42  B43  B53
                                          C32  B52  B62
                                               C42  C43
                                               D32  C52
                                                    D42
                                                    E32
The a(5) = 1 through a(15) = 14 partitions of n + 1 into three distinct parts [Uslu and Becenen] are the following (A = 10, B = 11, C = 12, D = 13). The ordered version is A211540(n)*6 and the Heinz numbers are A007304.
  321  421  431  432  532  542  543  643  653  654  754
            521  531  541  632  642  652  743  753  763
                 621  631  641  651  742  752  762  853
                      721  731  732  751  761  843  862
                           821  741  832  842  852  871
                                831  841  851  861  943
                                921  931  932  942  952
                                     A21  941  951  961
                                          A31  A32  A42
                                          B21  A41  A51
                                               B31  B32
                                               C21  B41
                                                    C31
                                                    D21
(End)

Robert Israel, Table of n, a(n) for n = 0..10000
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
Ece Uslu and Esin Becenen, Identical Object Distributions.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A001399, A069905, A211422.
All of the following pertain to 3-part strict partitions.
- A000009 counts these partitions of any length, with non-strict version A000041.
- A007304 gives the Heinz numbers, with non-strict version A014612.
- A101271 counts the relatively prime case, with non-strict version A023023.
- A220377 counts the pairwise coprime case, with non-strict version A307719.
- A337605 counts the pairwise non-coprime case, with non-strict version A337599.
Cf. A000217, A001840, A156040, A284825, A337453, A337483, A337484, A337563.

I:=[0,0,0,0,0,1]; [n le 6 select I[n] else Self(n-1)+Self(n-2)-Self(n-4)-Self(n-5)+Self(n-6): n in [1..70]]; // Vincenzo Librandi, Dec 31 2015

f:= gfun:-rectoproc({a(n) = a(n-1)+a(n-2)-a(n-4)-a(n-5)+a(n-6),seq(a(i)=0,i=0..4),a(5)=1},a(n),remember):
seq(f(i),i=0..100); # Robert Israel, Dec 31 2015

t[n_] := t[n] = Flatten[Table[-2 w + 3 x + 4 y, {w, n}, {x, n}, {y, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 80}]  (* A211540 *)
FindLinearRecurrence[t]
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 0, 0, 0, 0, 1}, 70] (* Vincenzo Librandi, Dec 31 2015 *)
Table[Length[Select[IntegerPartitions[n+1,{3}],UnsameQ@@#&]],{n,0,30}] (* Gus Wiseman, Oct 05 2020 *)

{a(n) = round( (n-2)^2 / 12 )}; / * Michael Somos, Feb 02 2015 */

concat(vector(5), Vec(x^5/(1-x-x^2+x^4+x^5-x^6) + O(x^100))) \\ Altug Alkan, Jan 10 2016

a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6).
a(n) = A069905(n-2) = A001399(n-5) for n >= 5. - Alois P. Heinz, Nov 03 2012
a(n) = 3*k^2-6*k+3 (for n = 6*k-3), 3*k^2-5*k+2 (for n = 6*k-2), 3*k^2-4*k+1 (for n = 6*k-1), 3*k^2-3*k+1 (for n = 6*k), 3*k^2-2*k (for n = 6*k+1), 3*k^2-k (for n = 6*k+2). - Ece Uslu, Esin Becenen, Dec 31 2015
a(n) = A004526(n-2) + a(n-2) for n > 2. - Ece Uslu, Esin Becenen, Dec 31 2015
G.f.: x^5/(1 - x - x^2 + x^4 + x^5 - x^6). - Robert Israel, Dec 31 2015
a(n) = Sum_{k=1..floor(n/3)} floor((n-k)/2)-k. - Wesley Ivan Hurt, Apr 27 2019
From Gus Wiseman, Oct 11 2020: (Start)
a(n+2) = A069905(n) = A001399(n-3) counts 3-part partitions.
a(n-1) = A069905(n-3) = A001399(n-6) counts 3-part strict partitions.
a(n-1) = A069905(n-3) = A001399(n-6) counts 3-part partitions with no 1's.
a(n-4) = A069905(n-6) = A001399(n-9) counts 3-part strict partitions with no 1's.
A000217(n-2) counts 3-part compositions.
a(n-1)*6 = A069905(n-3)*6 = A001399(n-6)*6 counts 3-part strict compositions.
A000217(n-5) counts 3-part compositions with no 1's.
a(n-4)*6 = A069905(n-6)*6 = A001399(n-9)*6 counts 3-part strict compositions with no 1's.
(End)

A284825 Number of partitions of n into 3 parts without common divisors such that every pair of them has common divisors.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 3, 0, 5, 0, 0, 0, 1, 0, 5, 0, 1, 0, 6, 0, 6, 0, 0, 0, 4, 0, 6, 0, 0, 0, 9, 0, 2, 1, 2, 0, 9, 0, 8, 1, 1, 0, 5, 0, 14, 0, 1, 0, 15, 0, 14, 0, 0, 1, 14, 0, 14, 0, 2, 0, 15, 0, 6, 1, 2, 1, 11, 0, 18, 1, 1, 0, 10, 0, 23

Offset: 31

Alois P. Heinz, Apr 03 2017

The Heinz numbers of these partitions are the intersection of A014612 (triples), A289509 (relatively prime), and A337694 (pairwise non-coprime). - Gus Wiseman, Oct 16 2020

			a(31) = 1: [6,10,15] = [2*3,2*5,3*5].
a(41) = 2: [6,14,21], [6,15,20].
From _Gus Wiseman_, Oct 14 2020: (Start)
Selected terms and the corresponding triples:
  a(31)=1: a(41)=2: a(59)=3:  a(77)=4:  a(61)=5:  a(71)=6:
-------------------------------------------------------------
  15,10,6  20,15,6  24,20,15  39,26,12  33,22,6   39,26,6
           21,14,6  24,21,14  42,20,15  40,15,6   45,20,6
                    35,14,10  45,20,12  45,10,6   50,15,6
                              50,15,12  28,21,12  35,21,15
                                        36,15,10  36,20,15
                                                  36,21,14
(End)

Alois P. Heinz, Table of n, a(n) for n = 31..10000

Cf. A082024, A230034, A230035.
A023023 does not require pairwise non-coprimality, with strict case A101271.
A202425 and A328672 count these partitions of any length, ranked by A328868.
A284825*6 is the ordered version.
A307719 is the pairwise coprime instead of non-coprime version.
A337599 does not require relatively primality, with strict case A337605.
A200976 and A328673 count pairwise non-coprime partitions.
A289509 gives Heinz numbers of relatively prime partitions.
A327516 counts pairwise coprime partitions, ranked by A333227.
A337694 gives Heinz numbers of pairwise non-coprime partitions.
Cf. A000217, A000741, A001399, A007304, A014612, A220377, A318716, A328679, A337604, A337666.

a:= proc(n) option remember; add(add(`if`(igcd(i, j)>1
      and igcd(i, j, n-i-j)=1 and igcd(i, n-i-j)>1 and
      igcd(j, n-i-j)>1, 1, 0), j=i..(n-i)/2), i=2..n/3)
    end:
seq(a(n), n=31..137);

a[n_] := a[n] = Sum[Sum[If[GCD[i, j] > 1 && GCD[i, j, n - i - j] == 1 && GCD[i, n - i - j] > 1 && GCD[j, n - i - j] > 1, 1, 0], {j, i, (n - i)/2} ], {i, 2, n/3}];
Table[a[n], {n, 31, 137}] (* Jean-François Alcover, Jun 13 2018, from Maple *)
stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
Table[Length[Select[IntegerPartitions[n,{3}],GCD@@#==1&&stabQ[#,CoprimeQ]&]],{n,31,100}] (* Gus Wiseman, Oct 14 2020 *)

a(n) > 0 iff n in { A230035 }.

a(n) = 0 iff n in { A230034 }.

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

A337605 Number of unordered triples of distinct positive integers summing to n, any two of which have a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 4, 0, 4, 1, 5, 0, 9, 0, 8, 3, 10, 0, 17, 1, 14, 5, 16, 1, 25, 1, 21, 8, 26, 2, 37, 1, 30, 15, 33, 2, 49, 2, 44, 16, 44, 2, 64, 6, 54, 21, 56, 3, 87, 5, 65, 30, 70, 9, 101, 5, 80, 34, 98, 6, 121, 6, 96, 52

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

			The a(n) triples for n = 12, 16, 18, 22, 27, 55:
  (6,4,2)  (8,6,2)   (8,6,4)   (10,8,4)  (12,9,6)  (28,21,6)
           (10,4,2)  (9,6,3)   (12,6,4)  (15,9,3)  (30,20,5)
                     (10,6,2)  (12,8,2)  (18,6,3)  (35,15,5)
                     (12,4,2)  (14,6,2)            (40,10,5)
                               (16,4,2)            (25,20,10)
                                                   (30,15,10)

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

A014612 intersected with A318719 ranks these partitions.
A220377 is the coprime instead of non-coprime version.
A318717 counts these partitions of any length, ranked by A318719.
A337599 is the non-strict version.
A337604 is the ordered non-strict version.
A337605*6 is the ordered version.
A023023 counts relatively prime 3-part partitions
A051424 counts pairwise coprime or singleton partitions.
A200976 and A328673 count pairwise non-coprime partitions.
A307719 counts pairwise coprime 3-part partitions.
A327516 counts pairwise coprime partitions, with strict case A305713.
Cf. A000217, A001399, A014612, A082024, A178472, A220377, A284825, A337461, A337561, A337667.

stabQ[u_,Q_]:=Array[#1==#2||!Q[u[[#1]],u[[#2]]]&,{Length[u],Length[u]},1,And];
Table[Length[Select[IntegerPartitions[n,{3}],UnsameQ@@#&&stabQ[#,CoprimeQ]&]],{n,0,100}]

A000741 Number of compositions of n into 3 ordered relatively prime parts.

Original entry on oeis.org

0, 0, 1, 3, 6, 9, 15, 18, 27, 30, 45, 42, 66, 63, 84, 84, 120, 99, 153, 132, 174, 165, 231, 180, 270, 234, 297, 270, 378, 276, 435, 360, 450, 408, 540, 414, 630, 513, 636, 552, 780, 558, 861, 690, 828, 759, 1035, 744, 1113, 870, 1104, 972, 1326, 945, 1380, 1116, 1386, 1218

Offset: 1

N. J. A. Sloane

			From _Gus Wiseman_, Oct 14 2020: (Start)
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)  (1,2,4)  (1,2,5)
           (2,1,1)  (1,3,1)  (1,3,2)  (1,3,3)  (1,3,4)
                    (2,1,2)  (1,4,1)  (1,4,2)  (1,4,3)
                    (2,2,1)  (2,1,3)  (1,5,1)  (1,5,2)
                    (3,1,1)  (2,3,1)  (2,1,4)  (1,6,1)
                             (3,1,2)  (2,2,3)  (2,1,5)
                             (3,2,1)  (2,3,2)  (2,3,3)
                             (4,1,1)  (2,4,1)  (2,5,1)
                                      (3,1,3)  (3,1,4)
                                      (3,2,2)  (3,2,3)
                                      (3,3,1)  (3,3,2)
                                      (4,1,2)  (3,4,1)
                                      (4,2,1)  (4,1,3)
                                      (5,1,1)  (4,3,1)
                                               (5,1,2)
                                               (5,2,1)
                                               (6,1,1)
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2(4) (1964), 241-260.
C. Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.
N. J. A. Sloane, Transforms

A000010 is the length-2 version.
A000217(n-2) does not require relative primality.
A000740 counts these compositions of any length.
A000742 is the length-4 version.
A000837 counts relatively prime partitions.
A023023 is the unordered version.
A101271 is the strict case.
A101391 has this as column k = 3.
A284825*6 is the pairwise non-coprime case.
A291166 intersected with A014311 ranks these compositions.
A337461 is the pairwise coprime instead of relatively prime version.
A337603 counts length-3 compositions whose distinct parts are pairwise coprime.
A337604 is the pairwise non-coprime instead of relatively prime version.
Cf. A001399, A007997, A023022, A078374, A337450, A337451, A337602.

with(numtheory):
mobtr:= proc(p)
          proc(n) option remember;
            add(mobius(n/d)*p(d), d=divisors(n))
          end
        end:
A000217:= n-> n*(n+1)/2:
a:= mobtr(n-> A000217(n-2)):
seq(a(n), n=1..58);  # Alois P. Heinz, Feb 08 2011

mobtr[p_] := Module[{f}, f[n_] := f[n] = Sum[MoebiusMu[n/d]*p[d], {d, Divisors[n]}]; f]; A000217[n_] := n*(n+1)/2; a = mobtr[A000217[#-2]&]; Table[a[n], {n, 1, 58}] (* Jean-François Alcover, Mar 12 2014, after Alois P. Heinz *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],GCD@@#==1&]],{n,0,30}] (* Gus Wiseman, Oct 14 2020 *)

Moebius transform of A000217(n-2).

G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = (1 - 3*x + 3*x^2)/(1 - x)^3. - Ilya Gutkovskiy, Apr 26 2017

Edited by Alois P. Heinz, Feb 08 2011

A337601 Number of unordered 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, 1, 2, 2, 3, 4, 4, 5, 6, 8, 7, 10, 7, 11, 11, 17, 12, 19, 12, 19, 17, 29, 16, 28, 19, 31, 23, 46, 23, 42, 25, 45, 27, 59, 31, 57, 34, 61, 37, 84, 38, 75, 42, 74, 47, 107, 45, 98, 51, 96, 56, 135, 54, 115, 63, 117, 67, 174, 65, 139, 75, 144, 75, 194

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

First differs from A337600 at a(9) = 4, A337600(9) = 5.

			The a(3) = 1 through a(14) = 10 partitions (A = 10, B = 11, C = 12):
  111  211  221  321  322  332  441  433  443  543  544  554
            311  411  331  431  522  532  533  552  553  743
                      511  521  531  541  551  651  661  752
                           611  711  721  722  732  733  761
                                     811  731  741  751  833
                                          911  831  922  851
                                               921  B11  941
                                               A11       A31
                                                         B21
                                                         C11

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

A014612 intersected with A304711 ranks these partitions.
A220377 is the strict case.
A304709 counts these partitions of any length.
A307719 is the strict case except for any number of 1's.
A337600 considers singletons to be coprime.
A337603 is the ordered version.
A000217 counts 3-part compositions.
A000837 counts relatively prime partitions.
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. A001840, A007359, A007360, A087087, A284825, A302696, A304712, A328673, A335238, A337602.

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

For n > 0, a(n) = A337600(n) - A079978(n).

A337450 Number of relatively prime compositions of n with no 1's.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 7, 5, 17, 17, 54, 51, 143, 168, 358, 482, 986, 1313, 2583, 3663, 6698, 9921, 17710, 26489, 46352, 70928, 121137, 188220, 317810, 497322, 832039, 1313501, 2177282, 3459041, 5702808, 9094377, 14930351, 23895672, 39084070, 62721578

Offset: 0

Gus Wiseman, Aug 31 2020

nonn

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

			The a(5) = 2 through a(10) = 17 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)    (2,3,3)  (5,4)      (2,3,5)
            (5,2)    (3,2,3)  (7,2)      (2,5,3)
            (2,2,3)  (3,3,2)  (2,2,5)    (3,2,5)
            (2,3,2)           (2,3,4)    (3,3,4)
            (3,2,2)           (2,4,3)    (3,4,3)
                              (2,5,2)    (3,5,2)
                              (3,2,4)    (4,3,3)
                              (3,4,2)    (5,2,3)
                              (4,2,3)    (5,3,2)
                              (4,3,2)    (2,2,3,3)
                              (5,2,2)    (2,3,2,3)
                              (2,2,2,3)  (2,3,3,2)
                              (2,2,3,2)  (3,2,2,3)
                              (2,3,2,2)  (3,2,3,2)
                              (3,2,2,2)  (3,3,2,2)

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 171 terms from Fausto A. C. Cariboni)

A000740 is the version allowing 1's.
2*A055684(n) is the case of length 2.
A302697 ranks the unordered case.
A302698 is the unordered version.
A337451 is the strict version.
A337452 is the unordered strict version.
A000837 counts relatively prime partitions.
A002865 counts partitions with no 1's.
A101268 counts singleton or pairwise coprime compositions.
A212804 counts compositions with no 1's.
A291166 appears to rank relatively prime compositions.
A337462 counts pairwise coprime compositions.
Cf. A000010, A007359, A023023, A101268, A178472, A289509, A302568, A337485.

b:= proc(n, g) option remember; `if`(n=0,
     `if`(g=1, 1, 0), add(b(n-j, igcd(g, j)), j=2..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..42);

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

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}]

A023035 Number of compositions of n into 10 ordered relatively prime parts.

Original entry on oeis.org

1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48620, 92377, 167960, 293920, 497420, 817135, 1307504, 2042755, 3124550, 4686110, 6906900, 10013002, 14307150, 20155070, 28048790, 38555660, 52451256, 70583095, 94143280, 124355000

Offset: 10

David W. Wilson

nonn

Marius A. Burtea, Table of n, a(n) for n = 10..5000
N. J. A. Sloane, Transforms

Cf. A000741, A000742, A000743, A023031, A023032, A023033, A023034.

Cf. A023023, A023024, A023025, A023026, A023027, A023028, A023029, A023030.

[&+[MoebiusMu(n div d)*Binomial(d-1, 9):d in Divisors(n)]:n in[10..42]]; // Marius A. Burtea, Feb 07 2020

with(numtheory):
a:= n-> add(mobius(n/d)*binomial(d-1, 9), d=divisors(n)):
seq(a(n), n=10..50);  # Alois P. Heinz, Feb 05 2020

a[n_]:=Sum[Boole[Divisible[n, k]] MoebiusMu[n/k] Binomial[k -1, 9], {k, 1, n}]; Table[a[n], {n, 10, 45}] (* or *) Table[a[n],{n, 9, 45}] a[n_]:=DivisorSum[n, Binomial[#-1,9] MoebiusMu[n/#]&]; Array[a, 37, 10] (* Vincenzo Librandi, Feb 08 2020 *)

Moebius transform of C(n-1,9).

G.f.: Sum_{k>=1} mu(k) * x^(10*k) / (1 - x^k)^10. - Ilya Gutkovskiy, Feb 05 2020

cp's OEIS Frontend

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

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A211540 Number of ordered triples (w,x,y) with all terms in {1..n} and 2w = 3x + 4y.

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A284825 Number of partitions of n into 3 parts without common divisors such that every pair of them has common divisors.

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

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A337605 Number of unordered triples of distinct positive integers summing to n, any two of which have a common divisor > 1.

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A000741 Number of compositions of n into 3 ordered relatively prime parts.

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A337601 Number of unordered 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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