A007360 -id:A007360 - cp's OEIS frontend

A335237 Numbers whose binary indices are not a singleton nor pairwise coprime.

0, 10, 11, 14, 15, 26, 27, 30, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 74, 75, 78, 79, 90, 91, 94, 95, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 114, 115, 116

Offset: 1

Gus Wiseman, May 28 2020

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The sequence of terms together with their binary expansions and binary indices begins:
    0:       0 ~ {}
   10:    1010 ~ {2,4}
   11:    1011 ~ {1,2,4}
   14:    1110 ~ {2,3,4}
   15:    1111 ~ {1,2,3,4}
   26:   11010 ~ {2,4,5}
   27:   11011 ~ {1,2,4,5}
   30:   11110 ~ {2,3,4,5}
   31:   11111 ~ {1,2,3,4,5}
   34:  100010 ~ {2,6}
   35:  100011 ~ {1,2,6}
   36:  100100 ~ {3,6}
   37:  100101 ~ {1,3,6}
   38:  100110 ~ {2,3,6}
   39:  100111 ~ {1,2,3,6}
   40:  101000 ~ {4,6}
   41:  101001 ~ {1,4,6}
   42:  101010 ~ {2,4,6}
   43:  101011 ~ {1,2,4,6}
   44:  101100 ~ {3,4,6}

The version for prime indices is A316438.
The version for standard compositions is A335236.
Numbers whose binary indices are pairwise coprime or a singleton: A087087.
Non-coprime partitions are counted by A335240.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Product is A124758.
- Reverse is A228351
- GCD is A326674.
- Heinz number is A333219.
- LCM is A333226.
Cf. A007360, A048793, A051424, A101268, A291166, A302569, A326675, A333227, A333228, A335235, A335239.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],!(Length[bpe[#]]==1||CoprimeQ@@bpe[#])&]

Complement in A001477 of A326675 and A000079.

A343654 Number of pairwise coprime sets of divisors > 1 of n.

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

Offset: 1

Gus Wiseman, Apr 26 2021

nonn

First differs from A100565 at a(210) = 52, A100565(210) = 51.

			The a(n) sets for n = 1, 2, 4, 6, 8, 12, 24, 30, 32, 36, 48:
  {}  {}   {}   {}     {}   {}     {}     {}       {}    {}     {}
      {2}  {2}  {2}    {2}  {2}    {2}    {2}      {2}   {2}    {2}
           {4}  {3}    {4}  {3}    {3}    {3}      {4}   {3}    {3}
                {6}    {8}  {4}    {4}    {5}      {8}   {4}    {4}
                {2,3}       {6}    {6}    {6}      {16}  {6}    {6}
                            {12}   {8}    {10}     {32}  {9}    {8}
                            {2,3}  {12}   {15}           {12}   {12}
                            {3,4}  {24}   {30}           {18}   {16}
                                   {2,3}  {2,3}          {36}   {24}
                                   {3,4}  {2,5}          {2,3}  {48}
                                   {3,8}  {3,5}          {2,9}  {2,3}
                                          {5,6}          {3,4}  {3,4}
                                          {2,15}         {4,9}  {3,8}
                                          {3,10}                {3,16}
                                          {2,3,5}

The version for partitions is A007359.
The version for subsets of {1..n} is A084422.
The case of pairs is A089233.
The version with 1's is A225520.
The maximal case is A343652.
The case without empty sets or singletons is A343653.
The maximal case without singletons is A343660.
A018892 counts pairwise coprime unordered pairs of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A100565 counts pairwise coprime unordered triples of divisors.
A187106, A276187, and A320426 count other types of pairwise coprime sets.
A326077 counts maximal pairwise indivisible sets.
Cf. A007360, A051026, A062319, A074206, A087087, A101268, A285572, A305713, A320423, A326675, A337485, A343655.

pwcop[y_]:=And@@(GCD@@#1==1&)/@Subsets[y,{2}];
Table[Length[Select[Subsets[Rest[Divisors[n]]],pwcop]],{n,100}]

A343659 Number of maximal pairwise coprime subsets of {1..n}.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 7, 9, 9, 10, 10, 12, 16, 19, 19, 20, 20, 22, 28, 32, 32, 33, 54, 61, 77, 84, 84, 85, 85, 94, 112, 123, 158, 161, 161, 176, 206, 212, 212, 214, 214, 229, 241, 260, 260, 263, 417, 428, 490, 521, 521, 526, 655, 674, 764, 818, 818, 820, 820, 874, 918, 975, 1182, 1189, 1189

Offset: 1

Gus Wiseman, Apr 26 2021

nonn

For this sequence, it does not matter whether singletons are considered pairwise coprime.
For n > 2, also the number of maximal pairwise coprime subsets of {2..n}.
For each prime p <= n, p divides exactly one element of each maximal subset. - Bert Dobbelaere, May 04 2021

			The a(1) = 1 through a(9) = 7 subsets:
  {1}  {12}  {123}  {123}  {1235}  {156}   {1567}   {1567}   {1567}
                    {134}  {1345}  {1235}  {12357}  {12357}  {12357}
                                   {1345}  {13457}  {13457}  {12579}
                                                    {13578}  {13457}
                                                             {13578}
                                                             {14579}
                                                             {15789}

Bert Dobbelaere, Table of n, a(n) for n = 1..500
Bert Dobbelaere, Python program

The case of pairs is A015614.
The case of triples is A015617.
The non-maximal version counting empty sets and singletons is A084422.
The non-maximal version counting singletons is A187106.
The non-maximal version is A320426(n) = A276187(n) + 1.
The version for indivisibility instead of coprimality is A326077.
The version for sets of divisors is A343652.
The version for sets of divisors > 1 is A343660.
A018892 counts coprime unordered pairs of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A100565 counts pairwise coprime unordered triples of divisors.
Cf. A007360, A067824, A087087, A225520, A324837, A325683, A325859, A326358, A326496, A326675, A333227, A343653, A343655.

fasmax[y_]:=Complement[y,Union@@Most@*Subsets/@y];
Table[Length[fasmax[Select[Subsets[Range[n]],CoprimeQ@@#&]]],{n,15}]

More terms from Bert Dobbelaere, May 04 2021

A337695 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are not pairwise coprime, where a singleton is always considered coprime.

Original entry on oeis.org

34, 40, 69, 70, 81, 88, 98, 104, 130, 138, 139, 141, 142, 160, 162, 163, 168, 177, 184, 197, 198, 209, 216, 226, 232, 260, 261, 262, 274, 276, 277, 278, 279, 282, 283, 285, 286, 288, 290, 296, 321, 324, 325, 326, 327, 328, 337, 344, 352, 354, 355, 360, 369

Offset: 1

Gus Wiseman, Sep 22 2020

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence together with the corresponding compositions begins:
     34: (4,2)        163: (2,4,1,1)    277: (4,2,2,1)
     40: (2,4)        168: (2,2,4)      278: (4,2,1,2)
     69: (4,2,1)      177: (2,1,4,1)    279: (4,2,1,1,1)
     70: (4,1,2)      184: (2,1,1,4)    282: (4,1,2,2)
     81: (2,4,1)      197: (1,4,2,1)    283: (4,1,2,1,1)
     88: (2,1,4)      198: (1,4,1,2)    285: (4,1,1,2,1)
     98: (1,4,2)      209: (1,2,4,1)    286: (4,1,1,1,2)
    104: (1,2,4)      216: (1,2,1,4)    288: (3,6)
    130: (6,2)        226: (1,1,4,2)    290: (3,4,2)
    138: (4,2,2)      232: (1,1,2,4)    296: (3,2,4)
    139: (4,2,1,1)    260: (6,3)        321: (2,6,1)
    141: (4,1,2,1)    261: (6,2,1)      324: (2,4,3)
    142: (4,1,1,2)    262: (6,1,2)      325: (2,4,2,1)
    160: (2,6)        274: (4,3,2)      326: (2,4,1,2)
    162: (2,4,2)      276: (4,2,3)      327: (2,4,1,1,1)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

A304712 counts the complement, with ordered version A337664.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A335238 does not consider a singleton coprime unless it is (1).
A337600 counts 3-part partitions in the complement.
A000740 counts relatively prime compositions.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A337461 counts pairwise coprime 3-part compositions.
A337561 counts pairwise coprime strict compositions.
A337665 counts compositions whose distinct parts are pairwise coprime.
A337666 ranks pairwise non-coprime compositions.
Cf. A007359, A007360, A087087, A302569, A302696, A304709, A305713, A335235, A337562, A337601, A337602, A337603.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!(SameQ@@stc[#]||CoprimeQ@@Union[stc[#]])&]

A343660 Number of maximal pairwise coprime sets of at least two divisors > 1 of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 4, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 4, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 8, 0, 1, 2, 0, 1, 4, 0, 2, 1, 4, 0, 6, 0, 1, 2, 2, 1, 4, 0, 4, 0, 1, 0, 8, 1, 1, 1

Offset: 1

Gus Wiseman, Apr 26 2021

nonn

			The a(n) sets for n = 6, 12, 24, 30, 36, 60, 72, 96:
  {2,3}  {2,3}  {2,3}  {5,6}    {2,3}  {5,6}    {2,3}  {2,3}
         {3,4}  {3,4}  {2,15}   {2,9}  {2,15}   {2,9}  {3,4}
                {3,8}  {3,10}   {3,4}  {3,10}   {3,4}  {3,8}
                       {2,3,5}  {4,9}  {3,20}   {3,8}  {3,16}
                                       {4,15}   {4,9}  {3,32}
                                       {5,12}   {8,9}
                                       {2,3,5}
                                       {3,4,5}

The case of pairs is A089233.
The case with 1's is A343652.
The case with singletons is (also) A343652.
The non-maximal version is A343653.
The non-maximal version with 1's is A343655.
The version for subsets of {2..n} is A343659 (for n > 2).
A018892 counts coprime unordered pairs of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A066620 counts pairwise coprime 3-sets of divisors.
A100565 counts pairwise coprime unordered triples of divisors.
Cf. A005361, A007359, A007360, A067824, A074206, A225520, A276187, A320426, A325683, A326077, A326359, A326496, A337485, A343654.

fasmax[y_]:=Complement[y,Union@@Most@*Subsets/@y];
Table[Length[fasmax[Select[Subsets[Rest[Divisors[n]]],CoprimeQ@@#&]]],{n,100}]

a(n) = A343652(n) - A005361(n).

A083290 Number of partitions of n into distinct parts which are coprime to n and which are also pairwise relatively prime.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 3, 2, 3, 2, 7, 2, 9, 3, 4, 5, 16, 3, 20, 4, 8, 7, 31, 5, 22, 9, 18, 9, 54, 4, 68, 16, 21, 16, 28, 11, 112, 20, 32, 18, 144, 9, 173, 22, 33, 40, 221, 19, 139, 25, 71, 43, 327, 25, 117, 47, 103, 80, 475, 18, 568, 90, 98, 122, 191, 29, 805, 93, 197, 44

Offset: 1

Reinhard Zumkeller, Apr 23 2003

nonn

a(n) <= A036998(n); see A082415 for numbers m with a(m) = A036998(m).

			a(7) = 3 since 7 = 3+4 = 2+5 = 1+6; 7 = 1+2+4 does not count (A036998(7)=4).

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000010, A000586, A007360.

a[n_] := a[n] = If[n == 1, 1, Module[{ip}, ip = IntegerPartitions[n, All, Select[Range[n - 1], CoprimeQ[#, n] &]]; Length@Select[ip, Sort[#] == Union[#] && AllTrue[Subsets[#, {2}], CoprimeQ @@ # &] &]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 80}] (* Jean-François Alcover, Dec 12 2021 *)

a(n)={local(Cache=Map()); my(recurse(r,p,k)=my(hk=[r,p,k],z); if(!mapisdefined(Cache,hk,&z), z=if(k==0, r==0, self()(r,p,k-1) + if(gcd(p,k)==1, self()(r-k, p*k, min(r-k,k-1)))); mapput(Cache, hk, z)); z); recurse(n,n,n)} \\ Andrew Howroyd, Apr 20 2021

Terms a(31) and beyond from Andrew Howroyd, Apr 20 2021

A184956 Maximum number of parts in a partition of n into pairwise coprime parts that are >= 2.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 4, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 5, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 6, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 7, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 8, 8, 9

Offset: 2

Max Alekseyev, Mar 17 2011

nonn

Fausto A. C. Cariboni, Table of n, a(n) for n = 2..700

Cf. A000837, A007359, A007360, A051424, A187718.

A187718 Number of partitions of n into maximum number (=A184956(n)) of pairwise coprime parts that are >= 2.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 4, 2, 5, 2, 1, 4, 1, 7, 1, 7, 1, 9, 5, 15, 5, 1, 6, 1, 12, 2, 14, 1, 14, 4, 23, 7, 30, 5, 1, 11, 1, 17, 1, 15, 1, 23, 5, 34, 4, 34, 5, 45, 12, 66, 11, 1, 13, 1, 25, 2, 25, 1, 27, 4, 46, 6, 54, 3, 56, 10, 90, 13, 101, 10, 1, 20, 1, 28, 1, 25, 1, 38, 4, 57, 2, 52, 4, 74, 9, 107, 6, 97, 9, 134, 18, 181, 17, 1, 19, 1

Offset: 1

Max Alekseyev, Mar 17 2011

nonn

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..700

Cf. A000837, A007359, A007360, A051424, A184956.

A282748 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_i, x_j) = 1 for all i != j (where 1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 3, 4, 1, 1, 2, 9, 4, 5, 1, 1, 6, 3, 16, 5, 6, 1, 1, 4, 15, 4, 25, 6, 7, 1, 1, 6, 9, 28, 5, 36, 7, 8, 1, 1, 4, 21, 16, 45, 6, 49, 8, 9, 1, 1, 10, 9, 52, 25, 66, 7, 64, 9, 10, 1, 1, 4, 39, 16, 105, 36, 91, 8, 81, 10, 11, 1, 1, 12, 9, 100, 25, 186, 49, 120, 9, 100, 11, 12, 1, 1, 6, 45, 16, 205, 36, 301, 64, 153, 10, 121, 12, 13, 1

Offset: 1

N. J. A. Sloane, Mar 05 2017

See A101391 for the triangle T(n,k) = number of compositions of n into k parts x_1, x_2, ..., x_k such that gcd(x_1,x_2,...,x_k) = 1 (2 <= k <= n).

			Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  2,  3,   1;
  1,  4,  3,   4,   1;
  1,  2,  9,   4,   5,   1;
  1,  6,  3,  16,   5,   6,  1;
  1,  4, 15,   4,  25,   6,  7,   1;
  1,  6,  9,  28,   5,  36,  7,   8,  1;
  1,  4, 21,  16,  45,   6, 49,   8,  9,   1;
  1, 10,  9,  52,  25,  66,  7,  64,  9,  10,  1;
  1,  4, 39,  16, 105,  36, 91,   8, 81,  10, 11,  1;
  1, 12,  9, 100,  25, 186, 49, 120,  9, 100, 11, 12, 1;
  ...
From _Gus Wiseman_, Nov 12 2020: (Start)
Row n = 6 counts the following compositions:
  (6)  (15)  (114)  (1113)  (11112)  (111111)
       (51)  (123)  (1131)  (11121)
             (132)  (1311)  (11211)
             (141)  (3111)  (12111)
             (213)          (21111)
             (231)
             (312)
             (321)
             (411)
(End)

Alois P. Heinz, Rows n = 1..200, flattened (first 100 rows from Chai Wah Wu)
Temba Shonhiwa, Compositions with pairwise relatively prime summands within a restricted setting, Fibonacci Quart. 44 (2006), no. 4, 316-323.

A072704 counts the unimodal instead of coprime version.
A087087 and A335235 rank these compositions.
A101268 gives row sums.
A101391 is the relatively prime instead of pairwise coprime version.
A282749 is the unordered version.
A000740 counts relatively prime compositions, with strict case A332004.
A007360 counts pairwise coprime or singleton strict partitions.
A051424 counts pairwise coprime or singleton partitions, ranked by A302569.
A097805 counts compositions by sum and length.
A178472 counts compositions with a common divisor.
A216652 and A072574 count strict compositions by sum and length.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions, ranked by A302696.
A335235 ranks pairwise coprime or singleton compositions.
A337462 counts pairwise coprime compositions, ranked by A333227.
A337562 counts pairwise coprime or singleton strict compositions.
A337665 counts compositions whose distinct parts are pairwise coprime, ranked by A333228.
Cf. A000837, A007359, A302568, A337461, A337561, A337667.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],Length[#]==1||CoprimeQ@@#&]],{n,10},{k,n}] (* Gus Wiseman, Nov 12 2020 *)

It seems that no general formula or recurrence is known, although Shonhiwa gives formulas for a few of the early diagonals.

A338317 Number of integer partitions of n with no 1's and pairwise coprime distinct parts, where a singleton is always considered coprime.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 5, 6, 7, 11, 11, 16, 16, 19, 25, 32, 34, 44, 46, 53, 66, 80, 88, 101, 116, 132, 150, 180, 204, 229, 254, 287, 331, 366, 426, 473, 525, 584, 662, 742, 835, 922, 1013, 1128, 1262, 1408, 1555, 1711, 1894, 2080, 2297, 2555, 2806, 3064, 3376

Offset: 0

Gus Wiseman, Oct 24 2020

nonn

			The a(2) = 1 through a(12) = 11 partitions (A = 10, B = 11, C = 12):
  2   3   4    5    6     7     8      9      A       B       C
          22   32   33    43    44     54     55      65      66
                    222   52    53     72     73      74      75
                          322   332    333    433     83      444
                                2222   522    532     92      543
                                       3222   3322    443     552
                                              22222   533     732
                                                      722     3333
                                                      3332    5322
                                                      5222    33222
                                                      32222   222222

A007359 (A302568) gives the strict case.
A101268 (A335235) gives pairwise coprime or singleton compositions.
A200976 (A338318) gives the pairwise non-coprime instead of coprime version.
A304709 (A304711) gives partitions whose distinct parts are pairwise coprime, with strict case A305713 (A302797).
A304712 (A338331) allows 1's, with strict version A007360 (A302798).
A327516 (A302696) gives pairwise coprime partitions.
A328673 (A328867) gives partitions with no distinct relatively prime parts.
A338315 (A337987) does not consider singletons coprime.
A338317 (A338316) gives these partitions.
A337462 (A333227) gives pairwise coprime compositions.
A337485 (A337984) gives pairwise coprime integer partitions with no 1's.
A337665 (A333228) gives compositions with pairwise coprime distinct parts.
A337667 (A337666) gives pairwise non-coprime compositions.
A337697 (A022340 /\ A333227) = pairwise coprime compositions with no 1's.
A337983 (A337696) gives pairwise non-coprime strict compositions, with unordered version A318717 (A318719).
Cf. A051424, A289508, A302569, A303140, A337694.

Table[Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&(SameQ@@#||CoprimeQ@@Union[#])&]],{n,0,15}]

The Heinz numbers of these partitions are given by A338316. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

cp's OEIS Frontend

A335237 Numbers whose binary indices are not a singleton nor pairwise coprime.

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A343654 Number of pairwise coprime sets of divisors > 1 of n.

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A343659 Number of maximal pairwise coprime subsets of {1..n}.

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A337695 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are not pairwise coprime, where a singleton is always considered coprime.

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A343660 Number of maximal pairwise coprime sets of at least two divisors > 1 of n.

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A083290 Number of partitions of n into distinct parts which are coprime to n and which are also pairwise relatively prime.

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A184956 Maximum number of parts in a partition of n into pairwise coprime parts that are >= 2.

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A187718 Number of partitions of n into maximum number (=A184956(n)) of pairwise coprime parts that are >= 2.

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A282748 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_i, x_j) = 1 for all i != j (where 1 <= k <= n).

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