A007359 -id:A007359 - cp's OEIS frontend

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

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}]

A366842 Number of integer partitions of n whose odd parts have a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 4, 1, 8, 3, 13, 6, 21, 10, 36, 15, 53, 28, 80, 41, 122, 63, 174, 97, 250, 140, 359, 201, 496, 299, 685, 410, 949, 575, 1284, 804, 1726, 1093, 2327, 1482, 3076, 2023, 4060, 2684, 5358, 3572, 6970, 4745, 9050, 6221, 11734, 8115, 15060, 10609

Offset: 0

Gus Wiseman, Oct 28 2023

nonn

			The a(3) = 1 through a(11) = 13 partitions:
  (3)  .  (5)    (3,3)  (7)      (3,3,2)  (9)        (5,5)      (11)
          (3,2)         (4,3)             (5,4)      (4,3,3)    (6,5)
                        (5,2)             (6,3)      (3,3,2,2)  (7,4)
                        (3,2,2)           (7,2)                 (8,3)
                                          (3,3,3)               (9,2)
                                          (4,3,2)               (4,4,3)
                                          (5,2,2)               (5,4,2)
                                          (3,2,2,2)             (6,3,2)
                                                                (7,2,2)
                                                                (3,3,3,2)
                                                                (4,3,2,2)
                                                                (5,2,2,2)
                                                                (3,2,2,2,2)

This is the odd case of A018783, complement A000837.
The even version is A047967.
The complement is counted by A366850, ranks A366846.
A000041 counts integer partitions, strict A000009.
A000740 counts relatively prime compositions.
A113685 counts partitions by sum of odds, stat A366528, w/o zeros A365067.
A168532 counts partitions by gcd.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A289508 gives gcd of prime indices, positions of ones A289509.
Cf. A007359, A051424, A055922, A066208, A078374, A087436, A116598, A337485, A366843, A366844, A366845.

Table[Length[Select[IntegerPartitions[n], GCD@@Select[#,OddQ]>1&]], {n,0,30}]

from math import gcd
from sympy.utilities.iterables import partitions
def A366842(n): return sum(1 for p in partitions(n) if gcd(*(q for q in p if q&1))>1) # Chai Wah Wu, Oct 28 2023

A366843 Number of integer partitions of n into odd, relatively prime parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 4, 6, 6, 9, 11, 13, 17, 21, 23, 32, 37, 42, 53, 62, 70, 88, 103, 116, 139, 164, 184, 220, 255, 283, 339, 390, 435, 511, 578, 653, 759, 863, 963, 1107, 1259, 1401, 1609, 1814, 2015, 2303, 2589, 2878, 3259, 3648, 4058, 4580, 5119, 5672, 6364

Offset: 0

Gus Wiseman, Oct 28 2023

nonn

			The a(1) = 1 through a(8) = 6 partitions:
  (1)  (11)  (111)  (31)    (311)    (51)      (331)      (53)
                    (1111)  (11111)  (3111)    (511)      (71)
                                     (111111)  (31111)    (3311)
                                               (1111111)  (5111)
                                                          (311111)
                                                          (11111111)

Allowing even parts gives A000837.
The strict case is A366844, with evens A078374.
The complement is counted by A366852, with evens A018783.
The pairwise coprime version is A366853, with evens A051424.
A000041 counts integer partitions, strict A000009 (also into odds).
A000740 counts relatively prime compositions.
A168532 counts partitions by gcd.
A366842 counts partitions whose odd parts have a common divisor > 1.
Cf. A007359, A047967, A055922, A066208, A113685, A116598, A289509, A289508, A302697, A337485, A366845, A366848, A366849.

Table[Length[Select[IntegerPartitions[n],#=={}||And@@OddQ/@#&&GCD@@#==1&]],{n,0,30}]

from math import gcd
from sympy.utilities.iterables import partitions
def A366843(n): return sum(1 for p in partitions(n) if all(d&1 for d in p) and gcd(*p)==1) # Chai Wah Wu, Oct 30 2023

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}]

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}]

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}]

A101391 Triangle read by rows: T(n,k) is the number of compositions of n into k parts x_1, x_2, ..., x_k such that gcd(x_1,x_2,...,x_k) = 1 (1<=k<=n).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 2, 3, 1, 0, 4, 6, 4, 1, 0, 2, 9, 10, 5, 1, 0, 6, 15, 20, 15, 6, 1, 0, 4, 18, 34, 35, 21, 7, 1, 0, 6, 27, 56, 70, 56, 28, 8, 1, 0, 4, 30, 80, 125, 126, 84, 36, 9, 1, 0, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 0, 4, 42, 154, 325, 461, 462, 330, 165, 55, 11, 1, 0, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1

Offset: 1

Emeric Deutsch, Jan 26 2005

If instead we require that the individual parts (x_i,x_j) be relatively prime, we get A282748. This is the question studied by Shonhiwa (2006). - N. J. A. Sloane, Mar 05 2017.

			T(6,3)=9 because we have 411,141,114 and the six permutations of 123 (222 does not qualify).
T(8,3)=18 because binomial(0,2)*mobius(8/1)+binomial(1,2)*mobius(8/2)+binomial(3,2)*mobius(8/4)+binomial(7,2)*mobius(8/8)=0+0+(-3)+21=18.
Triangle begins:
   1;
   0,  1;
   0,  2,  1;
   0,  2,  3,   1;
   0,  4,  6,   4,   1;
   0,  2,  9,  10,   5,   1;
   0,  6, 15,  20,  15,   6,   1;
   0,  4, 18,  34,  35,  21,   7,   1;
   0,  6, 27,  56,  70,  56,  28,   8,   1;
   0,  4, 30,  80, 125, 126,  84,  36,   9,   1;
   0, 10, 45, 120, 210, 252, 210, 120,  45,  10,  1;
   0,  4, 42, 154, 325, 461, 462, 330, 165,  55, 11,  1;
   0, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1;
  ...
From _Gus Wiseman_, Oct 19 2020: (Start)
Row n = 6 counts the following compositions:
  (15)  (114)  (1113)  (11112)  (111111)
  (51)  (123)  (1122)  (11121)
        (132)  (1131)  (11211)
        (141)  (1212)  (12111)
        (213)  (1221)  (21111)
        (231)  (1311)
        (312)  (2112)
        (321)  (2121)
        (411)  (2211)
               (3111)
Missing are: (42), (24), (33), (222).
(End)

Alois P. Heinz, Rows n = 1..200, flattened
H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2(4) (1964), 241-260.
Temba Shonhiwa, Compositions with pairwise relatively prime summands within a restricted setting, Fibonacci Quart. 44 (2006), no. 4, 316-323.

Mirror image of A039911.
Row sums are A000740.
Columns 2-9 are: A000010, A000741, A000742, A000743, A023031, A023032, A023033, A023034.
A000837 counts relatively prime partitions.
A135278 counts compositions by length.
A282748 is the pairwise coprime instead of relatively prime version.
A282750 is the unordered version.
A291166 ranks these compositions (evidently).
T(2n+1,n+1) gives A000984.
Cf. A007359, A008683, A085411, A101268, A289508, A289509, A302697, A332004, A337485, A337563.

with(numtheory): T:=proc(n,k) local d, j, b: d:=divisors(n): for j from 1 to tau(n) do b[j]:=binomial(d[j]-1,k-1)*mobius(n/d[j]) od: sum(b[i],i=1..tau(n)) end: for n from 1 to 14 do seq(T(n,k),k=1..n) od; # yields the sequence in triangular form
# second Maple program:
b:= proc(n, g) option remember; `if`(n=0, `if`(g=1, 1, 0),
      expand(add(b(n-j, igcd(g, j))*x, j=1..n)))
    end:
T:= (n, k)-> coeff(b(n,0),x,k):
seq(seq(T(n,k), k=1..n), n=1..14);  # Alois P. Heinz, May 05 2025

t[n_, k_] := Sum[Binomial[d-1, k-1]*MoebiusMu[n/d], {d, Divisors[n]}]; Table[t[n, k], {n, 2, 14}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jan 20 2014 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],GCD@@#==1&]],{n,10},{k,2,n}] (* change {k,2,n} to {k,1,n} for the version with zeros. - Gus Wiseman, Oct 19 2020 *)

T(n, k) = sumdiv(n, d, binomial(d-1, k-1)*moebius(n/d)); \\ Michel Marcus, Mar 09 2016

T(n,k) = Sum_{d|n} binomial(d-1,k-1)*mobius(n/d).

Sum_{k=1..n} k * T(n,k) = A085411(n). - Alois P. Heinz, May 05 2025

Definition clarified by N. J. A. Sloane, Mar 05 2017

Edited by Alois P. Heinz, May 05 2025

A337600 Number of unordered 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, 1, 2, 3, 3, 4, 5, 5, 6, 9, 7, 10, 8, 11, 11, 18, 12, 19, 13, 19, 17, 30, 16, 28, 20, 31, 23, 47, 23, 42, 26, 45, 27, 60, 31, 57, 35, 61, 37, 85, 38, 75, 43, 74, 47, 108, 45, 98, 52, 96, 56, 136, 54, 115, 64, 117, 67, 175, 65, 139, 76, 144, 75, 195

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

First differs from A337601 at a(9) = 5, A337601(9) = 4.

			The a(3) = 1 through a(14) = 10 partitions (A = 10, B = 11, C = 12):
  111  211  221  222  322  332  333  433  443  444  544  554
            311  321  331  431  441  532  533  543  553  743
                 411  511  521  522  541  551  552  661  752
                           611  531  721  722  651  733  761
                                711  811  731  732  751  833
                                          911  741  922  851
                                               831  B11  941
                                               921       A31
                                               A11       B21
                                                         C11

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

A220377 is the strict case.
A304712 counts these partitions of any length.
A307719 is the strict case except for any number of 1's.
A337601 does not consider a singleton to be coprime unless it is (1).
A337602 is the ordered version.
A337664 counts compositions of this type and any length.
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.
A304709 counts partitions whose distinct parts are pairwise coprime.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A337461 counts pairwise coprime length-3 compositions.
A337563 counts pairwise coprime length-3 partitions with no 1's.
Cf. A001840, A007359, A007360, A014612, A087087, A284825, A302569, A302696, A328673, A335235, A337603, A337695.

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

For n > 0, a(n) = A337601(n) + A079978(n).

A337664 Number of compositions of n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 30, 58, 111, 210, 396, 750, 1420, 2688, 5079, 9586, 18092, 34157, 64516, 121899, 230373, 435463, 823379, 1557421, 2946938, 5578111, 10561990, 20005129, 37902514, 71832373, 136173273, 258211603, 489738627, 929074448, 1762899110, 3345713034

Offset: 0

Gus Wiseman, Sep 21 2020

nonn

			The a(0) = 1 through a(5) = 16 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (211)   (122)
                        (1111)  (131)
                                (212)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..220

A304712 is the unordered version.
A337562 is the strict case.
A337602 is the length-3 case.
A337665 does not consider a singleton to be coprime unless it is (1).
A337695 ranks the complement of these compositions.
A000740 counts relatively prime compositions.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A337461 counts pairwise coprime length-3 compositions.
A337561 counts pairwise coprime strict compositions.
Cf. A007359, A007360, A087087, A302569, A304709, A307719, A335235, A335238, A335239, A337562, A337603, A337667.

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

A366844 Number of strict integer partitions of n into odd relatively prime parts.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 0, 2, 1, 2, 1, 2, 2, 3, 3, 5, 4, 4, 5, 6, 7, 8, 8, 9, 11, 12, 12, 15, 16, 15, 19, 23, 23, 26, 28, 30, 34, 37, 38, 44, 48, 48, 56, 62, 63, 72, 77, 82, 92, 96, 102, 116, 124, 128, 142, 155, 162, 178, 191, 200, 222, 236, 246, 276, 291, 303, 334

Offset: 0

Gus Wiseman, Oct 29 2023

nonn

			The a(n) partitions for n = 1, 8, 14, 17, 16, 20, 21:
  (1)  (5,3)  (9,5)   (9,5,3)   (9,7)      (11,9)      (9,7,5)
       (7,1)  (11,3)  (9,7,1)   (11,5)     (13,7)      (11,7,3)
              (13,1)  (11,5,1)  (13,3)     (17,3)      (11,9,1)
                      (13,3,1)  (15,1)     (19,1)      (13,5,3)
                                (7,5,3,1)  (9,7,3,1)   (13,7,1)
                                           (11,5,3,1)  (15,5,1)
                                                       (17,3,1)

This is the relatively prime case of A000700.
The pairwise coprime version is the odd-part case of A007360.
Allowing even parts gives A078374.
The halved even version is A078374 aerated.
The non-strict version is A366843, with evens A000837.
The complement is counted by the strict case of A366852, with evens A018783.
A000041 counts integer partitions, strict A000009 (also into odds).
A051424 counts pairwise coprime partitions, for odd parts A366853.
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A168532 counts partitions by gcd.
A366842 counts partitions whose odd parts have a common divisor > 1.
Cf. A007359, A047967, A055922, A066208, A116598, A239261, A302697, A337485, A365067, A366845, A366848.

Table[Length[Select[IntegerPartitions[n], And@@OddQ/@#&&UnsameQ@@#&&GCD@@#==1&]],{n,0,30}]

from math import gcd
from sympy.utilities.iterables import partitions
def A366844(n): return sum(1 for p in partitions(n) if all(d==1 for d in p.values()) and all(d&1 for d in p) and gcd(*p)==1) # Chai Wah Wu, Oct 30 2023

More terms from Chai Wah Wu, Oct 30 2023

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

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

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

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A337600 Number of unordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.

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