A051424 -id:A051424 - cp's OEIS frontend

A337599 Number of unordered triples of positive integers summing to n, any two of which have a common divisor > 1.

0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 4, 0, 4, 3, 5, 0, 9, 0, 9, 5, 10, 0, 16, 2, 14, 7, 17, 0, 27, 1, 21, 11, 24, 6, 36, 1, 30, 15, 37, 2, 51, 1, 41, 25, 44, 2, 64, 5, 58, 25, 57, 2, 81, 13, 69, 31, 70, 3, 108, 5, 80, 43, 85, 17, 123, 5, 97, 46, 120, 6, 144, 6

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

First differs from A082024 at a(31) = 1, A082024(31) = 0.

The first relatively prime triple is (15,10,6), counted under a(31).

			The a(6) = 1 through a(16) = 5 partitions are (empty columns indicated by dots, A..G = 10..16):
  222  .  422  333  442  .  444  .  644  555  664  .  666  .  866
                    622     633     662  663  844     864     884
                            642     842  933  862     882     A55
                            822     A22       A42     963     A64
                                              C22     A44     A82
                                                      A62     C44
                                                      C33     C62
                                                      C42     E42
                                                      E22     G22

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

A014612 intersected with A337694 ranks these partitions.
A200976 and A328673 count these partitions of any length.
A284825 is the case that is also relatively prime.
A307719 is the pairwise coprime instead of non-coprime version.
A335402 gives the positions of zeros.
A337604 is the ordered version.
A337605 is the strict case.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A305713 counts strict pairwise coprime partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A000212, A000217, A001840, A018783, A082024, A211540, A220377, A337461.

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

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}]

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

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Oct 05 2020

nonn

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

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

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

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

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

A340608 The number of prime factors of n (A001222) is relatively prime to the maximum prime index of n (A061395).

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 19, 22, 23, 25, 27, 28, 29, 31, 32, 33, 34, 37, 40, 41, 42, 43, 44, 46, 47, 48, 51, 53, 55, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 76, 77, 79, 80, 82, 83, 85, 88, 89, 90, 93, 94, 97, 98, 99

Offset: 1

Gus Wiseman, Jan 27 2021

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
     2: {1}          22: {1,5}          44: {1,1,5}
     3: {2}          23: {9}            46: {1,9}
     4: {1,1}        25: {3,3}          47: {15}
     5: {3}          27: {2,2,2}        48: {1,1,1,1,2}
     7: {4}          28: {1,1,4}        51: {2,7}
     8: {1,1,1}      29: {10}           53: {16}
    10: {1,3}        31: {11}           55: {3,5}
    11: {5}          32: {1,1,1,1,1}    59: {17}
    12: {1,1,2}      33: {2,5}          60: {1,1,2,3}
    13: {6}          34: {1,7}          61: {18}
    15: {2,3}        37: {12}           62: {1,11}
    16: {1,1,1,1}    40: {1,1,1,3}      63: {2,2,4}
    17: {7}          41: {13}           64: {1,1,1,1,1,1}
    18: {1,2,2}      42: {1,2,4}        66: {1,2,5}
    19: {8}          43: {14}           67: {19}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Note: Heinz numbers are given in parentheses below.
These are the Heinz numbers of the partitions counted by A200750.
The case of equality is A047993 (A106529).
The divisible instead of coprime version is A168659 (A340609).
The dividing instead of coprime version is A168659 (A340610), with strict case A340828 (A340856).
A001222 counts prime factors.
A006141 counts partitions whose length equals their minimum (A324522).
A051424 counts singleton or pairwise coprime partitions (A302569).
A056239 adds up prime indices.
A061395 selects the maximum prime index.
A067538 counts partitions whose length divides their sum (A316413).
A067538 counts partitions whose maximum divides their sum (A326836).
A112798 lists the prime indices of each positive integer.
A259936 counts singleton or pairwise coprime factorizations.
A325134 = A001222 + A061395.
A326845 = A056239 * A061395.
A326849 counts partitions whose sum divides length times maximum (A326848).
A327516 counts pairwise coprime partitions (A302696).
Cf. A003114, A039900, A096401, A244990/A244991, A340606, A340653, A340691, A340787/A340788.

Select[Range[100],GCD[PrimeOmega[#],PrimePi[FactorInteger[#][[-1,1]]]]==1&]

A303140 Number of strict integer partitions of n with at least two but not all parts having a common divisor greater than 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 4, 2, 8, 7, 14, 14, 21, 18, 33, 32, 50, 54, 72, 67, 103, 110, 145, 155, 201, 196, 271, 293, 372, 400, 493, 512, 647, 704, 858, 924, 1115, 1167, 1436, 1560, 1854, 2022, 2368, 2510, 3005, 3255, 3804, 4144, 4792, 5116, 5989, 6514, 7486

Offset: 1

Gus Wiseman, Apr 19 2018

nonn

			The a(14) = 7 partitions are (932), (8321), (7421), (653), (6521), (6431), (5432).

Cf. A000837, A018783, A051424, A078374, A168532, A289508, A289509, A298748, A300486, A302569, A302696, A302796, A303138, A303139.

Table[Select[IntegerPartitions[n],UnsameQ@@#&&!CoprimeQ@@#&&GCD@@#===1&]//Length,{n,20}]

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.

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).

A302568 Odd numbers that are either prime or whose prime indices are pairwise coprime.

Original entry on oeis.org

3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 55, 59, 61, 67, 69, 71, 73, 77, 79, 83, 85, 89, 93, 95, 97, 101, 103, 107, 109, 113, 119, 123, 127, 131, 137, 139, 141, 143, 145, 149, 151, 155, 157, 161, 163, 165, 167, 173, 177, 179

Offset: 1

Gus Wiseman, Apr 10 2018

nonn

Also Heinz numbers of partitions with pairwise coprime parts all greater than 1 (A007359), where singletons are considered coprime. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      3: {2}       43: {14}      89: {24}      141: {2,15}
      5: {3}       47: {15}      93: {2,11}    143: {5,6}
      7: {4}       51: {2,7}     95: {3,8}     145: {3,10}
     11: {5}       53: {16}      97: {25}      149: {35}
     13: {6}       55: {3,5}    101: {26}      151: {36}
     15: {2,3}     59: {17}     103: {27}      155: {3,11}
     17: {7}       61: {18}     107: {28}      157: {37}
     19: {8}       67: {19}     109: {29}      161: {4,9}
     23: {9}       69: {2,9}    113: {30}      163: {38}
     29: {10}      71: {20}     119: {4,7}     165: {2,3,5}
     31: {11}      73: {21}     123: {2,13}    167: {39}
     33: {2,5}     77: {4,5}    127: {31}      173: {40}
     35: {3,4}     79: {22}     131: {32}      177: {2,17}
     37: {12}      83: {23}     137: {33}      179: {41}
     41: {13}      85: {3,7}    139: {34}      181: {42}
Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of multiset systems.
03: {{1}}
05: {{2}}
07: {{1,1}}
11: {{3}}
13: {{1,2}}
15: {{1},{2}}
17: {{4}}
19: {{1,1,1}}
23: {{2,2}}
29: {{1,3}}
31: {{5}}
33: {{1},{3}}
35: {{2},{1,1}}
37: {{1,1,2}}
41: {{6}}
43: {{1,4}}
47: {{2,3}}
51: {{1},{4}}
53: {{1,1,1,1}}

A005117 is a superset.
A007359 counts partitions with these Heinz numbers.
A302569 allows evens, with squarefree version A302798.
A337694 is the pairwise non-coprime instead of pairwise coprime version.
A337984 does not include the primes.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions, ranked by A302696.
A337462 counts pairwise coprime compositions, ranked by A333227.
A337561 counts pairwise coprime strict compositions.
A337667 counts pairwise non-coprime compositions, ranked by A337666.
A337697 counts pairwise coprime compositions with no 1's.
Cf. A005408, A051424, A056239, A087087, A112798, A200976, A302797, A303282, A304711, A335235, A338468.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1,400,2],Or[PrimeQ[#],CoprimeQ@@primeMS[#]]&]

Equals A065091 \/ A337984.

Equals A302569 /\ A005408.

Extended by Gus Wiseman, Oct 29 2020

A304717 Number of connected strict integer partitions of n with pairwise indivisible parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 3, 2, 4, 3, 5, 2, 5, 4, 6, 3, 7, 6, 9, 5, 9, 8, 13, 10, 15, 9, 15, 13, 18, 14, 22, 21, 26, 19, 29, 24, 36, 31, 40, 35, 45, 38, 54, 55, 59, 55, 70, 69, 84, 74, 89, 86, 107, 103, 119, 115, 143, 143, 159

Offset: 1

Gus Wiseman, May 17 2018

nonn

Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph.

			The a(34) = 13 connected strict integer partitions with pairwise indivisible parts are (34), (18,16), (20,14), (22,12), (24,10), (26,8), (28,6), (30,4), (14,12,8), (15,10,9), (20,8,6), (14,10,6,4), (15,9,6,4). Their corresponding multiset multisystems (see A112798, A302242) are the following.
         (34): {{1,7}}
       (30 4): {{1,2,3},{1,1}}
       (28 6): {{1,1,4},{1,2}}
       (26 8): {{1,6},{1,1,1}}
      (24 10): {{1,1,1,2},{1,3}}
      (22 12): {{1,5},{1,1,2}}
      (20 14): {{1,1,3},{1,4}}
     (20 8 6): {{1,1,3},{1,1,1},{1,2}}
      (18 16): {{1,2,2},{1,1,1,1}}
    (15 10 9): {{2,3},{1,3},{2,2}}
   (15 9 6 4): {{2,3},{2,2},{1,2},{1,1}}
    (14 12 8): {{1,4},{1,1,2},{1,1,1}}
  (14 10 6 4): {{1,4},{1,3},{1,2},{1,1}}

Cf. A000009, A003963, A006126, A048143, A051424, A054921, A076078, A259936, A281116, A285572, A285573, A286518, A286520, A293993, A302242, A303362, A304714, A304716.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c==={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[zsm[#]]===1&&Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]==={}&]],{n,30}]

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

cp's OEIS Frontend

A337599 Number of unordered triples of positive integers summing to n, any two of which have a common divisor > 1.

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

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A340608 The number of prime factors of n (A001222) is relatively prime to the maximum prime index of n (A061395).

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A303140 Number of strict integer partitions of n with at least two but not all parts having a common divisor greater than 1.

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

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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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A302568 Odd numbers that are either prime or whose prime indices are pairwise coprime.

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