A333227 -id:A333227 - cp's OEIS frontend

A228351 Triangle read by rows in which row n lists the compositions (ordered partitions) of n (see Comments lines for definition).

1, 2, 1, 1, 3, 1, 2, 2, 1, 1, 1, 1, 4, 1, 3, 2, 2, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 5, 1, 4, 2, 3, 1, 1, 3, 3, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 4, 1, 1, 3, 1, 2, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 5, 2, 4, 1, 1, 4

Offset: 1

Omar E. Pol, Aug 30 2013

The representation of the compositions (for fixed n) is as lists of parts, the order between individual compositions (for the same n) is (list-)reversed co-lexicographic. - Joerg Arndt, Sep 02 2013
Dropping the "(list-)reversed" in the comment above gives A228525.
The equivalent sequence for partitions is A026792.
This sequence lists (without repetitions) all finite compositions, in such a way that, if [P_1, ..., P_r] denotes the composition occupying the n-th position in the list, then (((2*n/2^(P_1)-1)/2^(P_2)-1)/...)/2^(P_r)-1 = 0. - Lorenzo Sauras Altuzarra, Jan 22 2020
The k-th composition in the list is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, and taking first differences. Reversing again gives A066099, which is described as the standard ordering. Both sequences define a bijective correspondence between nonnegative integers and integer compositions. - Gus Wiseman, Apr 01 2020
It follows from the previous comment that A000120(k) is the length of the k-th composition that is listed by this sequence (recall that A000120(k) is the number of 1's in the binary expansion of k). - Lorenzo Sauras Altuzarra, Sep 29 2020

			Illustration of initial terms:
-----------------------------------
n  j     Diagram     Composition j
-----------------------------------
.         _
1  1     |_|         1;
.         _ _
2  1     |_  |       2,
2  2     |_|_|       1, 1;
.         _ _ _
3  1     |_    |     3,
3  2     |_|_  |     1, 2,
3  3     |_  | |     2, 1,
3  4     |_|_|_|     1, 1, 1;
.         _ _ _ _
4  1     |_      |   4,
4  2     |_|_    |   1, 3,
4  3     |_  |   |   2, 2,
4  4     |_|_|_  |   1, 1, 2,
4  5     |_    | |   3, 1,
4  6     |_|_  | |   1, 2, 1,
4  7     |_  | | |   2, 1, 1,
4  8     |_|_|_|_|   1, 1, 1, 1;
.
Triangle begins:
[1];
[2],[1,1];
[3],[1,2],[2,1],[1,1,1];
[4],[1,3],[2,2],[1,1,2],[3,1],[1,2,1],[2,1,1],[1,1,1,1];
[5],[1,4],[2,3],[1,1,3],[3,2],[1,2,2],[2,1,2],[1,1,1,2],[4,1],[1,3,1],[2,2,1],[1,1,2,1],[3,1,1],[1,2,1,1],[2,1,1,1],[1,1,1,1,1];
...
For example [1,2] occupies the 5th position in the corresponding list of compositions and indeed (2*5/2^1-1)/2^2-1 = 0. - _Lorenzo Sauras Altuzarra_, Jan 22 2020
12 --binary expansion--> [1,1,0,0] --reverse--> [0,0,1,1] --positions of 1's--> [3,4] --prepend 0--> [0,3,4] --first differences--> [3,1]. - _Lorenzo Sauras Altuzarra_, Sep 29 2020

Peter Kagey, Table of n, a(n) for n = 1..10000
Mikhail Kurkov, Comments on A228351
Index entries for sequences that are related to compositions

Row n has length A001792(n-1). Row sums give A001787, n >= 1.
Cf. A000120 (binary weight), A001511, A006519, A011782, A026792, A065120.
A related ranking of finite sets is A048793/A272020.
All of the following consider the k-th row to be the k-th composition, ignoring the coarser grouping by sum.
- Indices of weakly increasing rows are A114994.
- Indices of weakly decreasing rows are A225620.
- Indices of strictly decreasing rows are A333255.
- Indices of strictly increasing rows are A333256.
- Indices of reversed interval rows A164894.
- Indices of interval rows are A246534.
- Indices of strict rows are A233564.
- Indices of constant rows are A272919.
- Indices of anti-run rows are A333489.
- Row k has A124767(k) runs and A333381(k) anti-runs.
- Row k has GCD A326674(k) and LCM A333226(k).
- Row k has Heinz number A333219(k).
Cf. A000120, A029931, A035327, A070939, A233249, A333217, A333218, A333220, A333227, A333627, A333628.
Equals A163510+1, termwise.
Cf. A124734 (increasing length, then lexicographic).
Cf. A296774 (increasing length, then reverse lexicographic).
Cf. A337243 (increasing length, then colexicographic).
Cf. A337259 (increasing length, then reverse colexicographic).
Cf. A296773 (decreasing length, then lexicographic).
Cf. A296772 (decreasing length, then reverse lexicographic).
Cf. A337260 (decreasing length, then colexicographic).
Cf. A108244 (decreasing length, then reverse colexicographic).
Cf. A228369 (lexicographic).
Cf. A066099 (reverse lexicographic).
Cf. A228525 (colexicographic).

a228351 n = a228351_list !! (n - 1)
a228351_list = concatMap a228351_row [1..]
a228351_row 0 = []
a228351_row n = a001511 n : a228351_row (n `div` 2^(a001511 n))
-- Peter Kagey, Jun 27 2016

# Program computing the sequence:
A228351 := proc(n) local c, k, L, N: L, N := [], [seq(2*r, r = 1 .. n)]: for k in N do c := 0: while k != 0 do if gcd(k, 2) = 2 then k := k/2: c := c+1: else L := [op(L), op(c)]: k := k-1: c := 0: fi: od: od: L[n]: end: # Lorenzo Sauras Altuzarra, Jan 22 2020
# Program computing the list of compositions:
List := proc(n) local c, k, L, M, N: L, M, N := [], [], [seq(2*r, r = 1 .. 2^n-1)]: for k in N do c := 0: while k != 0 do if gcd(k, 2) = 2 then k := k/2: c := c+1: else L := [op(L), c]: k := k-1: c := 0: fi: od: M := [op(M), L]: L := []: od: M: end: # Lorenzo Sauras Altuzarra, Jan 22 2020

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Differences[Prepend[bpe[n],0]],{n,0,30}] (* Gus Wiseman, Apr 01 2020 *)

from itertools import count, islice
def A228351_gen(): # generator of terms
    for n in count(1):
        k = n
        while k:
            yield (s:=(~k&k-1).bit_length()+1)
            k >>= s
A228351_list = list(islice(A228351_gen(),30)) # Chai Wah Wu, Jul 17 2023

A101268 Number of compositions of n into pairwise relatively prime parts.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 22, 38, 63, 101, 160, 254, 403, 635, 984, 1492, 2225, 3281, 4814, 7044, 10271, 14889, 21416, 30586, 43401, 61205, 85748, 119296, 164835, 226423, 309664, 422302, 574827, 781237, 1060182, 1436368, 1942589, 2622079, 3531152, 4742316, 6348411

Offset: 0

Vladeta Jovovic, Dec 18 2004

nonn

Here a singleton is always considered pairwise relatively prime. Compare to A337462. - Gus Wiseman, Oct 18 2020

			From _Gus Wiseman_, Oct 18 2020: (Start)
The a(1) = 1 through a(5) = 13 compositions:
  (1)  (2)   (3)    (4)     (5)
       (11)  (12)   (13)    (14)
             (21)   (31)    (23)
             (111)  (112)   (32)
                    (121)   (41)
                    (211)   (113)
                    (1111)  (131)
                            (311)
                            (1112)
                            (1121)
                            (1211)
                            (2111)
                            (11111)
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500 (terms 0..400 from Alois P. Heinz)
Temba Shonhiwa, Compositions with pairwise relatively prime summands within a restricted setting, Fibonacci Quart. 44 (2006), no. 4, 316-323.

Row sums of A282748.
A051424 is the unordered version, with strict case A007360.
A335235 ranks these compositions.
A337461 counts these compositions of length 3, with unordered version A307719 and unordered strict version A220377.
A337462 does not consider a singleton to be coprime unless it is (1), with strict version A337561.
A337562 is the strict case.
A337664 looks only at distinct parts, with non-constant version A337665.
A000740 counts relatively prime compositions, with strict case A332004.
A178472 counts compositions with a common factor.
Cf. A087087, A302569, A305713, A326675, A327516, A328673, A333227, A333228.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[#]<=1||CoprimeQ@@#&]],{n,0,10}] (* Gus Wiseman, Oct 18 2020 *)

It seems that no formula is known.

a(0)=1 prepended by Alois P. Heinz, Jun 14 2017

A327516 Number of integer partitions of n that are empty, (1), or have at least two parts and these parts are pairwise coprime.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 9, 11, 14, 17, 22, 26, 32, 37, 42, 50, 59, 69, 80, 91, 101, 115, 133, 152, 170, 190, 210, 235, 265, 300, 334, 366, 398, 441, 484, 541, 597, 648, 703, 770, 848, 935, 1022, 1102, 1184, 1281, 1406, 1534, 1661, 1789, 1916, 2062, 2244, 2435

Offset: 0

Gus Wiseman, Sep 19 2019

nonn

The Heinz numbers of these partitions are given by A302696.

Note that the definition excludes partitions with repeated parts other than 1 (cf. A038348, A304709).

			The a(1) = 1 through a(8) = 11 partitions:
  (1)  (11)  (21)   (31)    (32)     (51)      (43)       (53)
             (111)  (211)   (41)     (321)     (52)       (71)
                    (1111)  (311)    (411)     (61)       (431)
                            (2111)   (3111)    (511)      (521)
                            (11111)  (21111)   (3211)     (611)
                                     (111111)  (4111)     (5111)
                                               (31111)    (32111)
                                               (211111)   (41111)
                                               (1111111)  (311111)
                                                          (2111111)
                                                          (11111111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..750
Gus Wiseman, Sequences counting and encoding certain classes of multisets

Cf. A038348, A302569, A304709, A304711.
A000837 is the relatively prime instead of pairwise coprime version.
A051424 includes all singletons, with strict case A007360.
A101268 is the ordered version (with singletons).
A302696 ranks these partitions, with complement A335241.
A305713 is the strict case.
A307719 counts these partitions of length 3.
A018783 counts partitions with a common divisor.
A328673 counts pairwise non-coprime partitions.
Cf. A087087, A220377, A326675, A333227, A333228, A335238.

Table[Length[Select[IntegerPartitions[n],#=={}||CoprimeQ@@#&]],{n,0,30}]

For n > 1, a(n) = A051424(n) - 1. - Gus Wiseman, Sep 18 2020

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

Original entry on oeis.org

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

A333228 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are pairwise coprime, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Gus Wiseman, May 28 2020

nonn

First differs from A291166 in lacking 69, which corresponds to the composition (4,2,1).
We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).
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:
   1: (1)          21: (2,2,1)        39: (3,1,1,1)
   3: (1,1)        22: (2,1,2)        41: (2,3,1)
   5: (2,1)        23: (2,1,1,1)      43: (2,2,1,1)
   6: (1,2)        24: (1,4)          44: (2,1,3)
   7: (1,1,1)      25: (1,3,1)        45: (2,1,2,1)
   9: (3,1)        26: (1,2,2)        46: (2,1,1,2)
  11: (2,1,1)      27: (1,2,1,1)      47: (2,1,1,1,1)
  12: (1,3)        28: (1,1,3)        48: (1,5)
  13: (1,2,1)      29: (1,1,2,1)      49: (1,4,1)
  14: (1,1,2)      30: (1,1,1,2)      50: (1,3,2)
  15: (1,1,1,1)    31: (1,1,1,1,1)    51: (1,3,1,1)
  17: (4,1)        33: (5,1)          52: (1,2,3)
  18: (3,2)        35: (4,1,1)        53: (1,2,2,1)
  19: (3,1,1)      37: (3,2,1)        54: (1,2,1,2)
  20: (2,3)        38: (3,1,2)        55: (1,2,1,1,1)

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

Pairwise coprime or singleton partitions are A051424.
Coprime or singleton sets are ranked by A087087.
The version for relatively prime instead of coprime appears to be A291166.
Numbers whose binary indices are pairwise coprime are A326675.
Coprime partitions are counted by A327516.
Not ignoring repeated parts gives A333227.
The complement is A335238.
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.
- Number of distinct parts is A334028.
Cf. A007360, A048793, A101268, A233564, A272919, A291166, A302569, A335235, A335236, A335237, A335239.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,120],CoprimeQ@@Union[stc[#]]&]

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

Original entry on oeis.org

0, 0, 0, 1, 3, 3, 9, 3, 15, 9, 21, 9, 39, 9, 45, 21, 45, 21, 87, 21, 93, 39, 87, 39, 153, 39, 135, 63, 153, 57, 255, 51, 207, 93, 225, 93, 321, 81, 291, 135, 321, 105, 471, 105, 393, 183, 381, 147, 597, 147, 531, 213, 507, 183, 759, 207, 621, 273, 621, 231

Offset: 0

Gus Wiseman, Sep 02 2020

nonn

			The a(3) = 1 through a(9) = 9 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)  (1,1,6)  (1,1,7)
           (1,2,1)  (1,3,1)  (1,2,3)  (1,5,1)  (1,2,5)  (1,3,5)
           (2,1,1)  (3,1,1)  (1,3,2)  (5,1,1)  (1,3,4)  (1,5,3)
                             (1,4,1)           (1,4,3)  (1,7,1)
                             (2,1,3)           (1,5,2)  (3,1,5)
                             (2,3,1)           (1,6,1)  (3,5,1)
                             (3,1,2)           (2,1,5)  (5,1,3)
                             (3,2,1)           (2,5,1)  (5,3,1)
                             (4,1,1)           (3,1,4)  (7,1,1)
                                               (3,4,1)
                                               (4,1,3)
                                               (4,3,1)
                                               (5,1,2)
                                               (5,2,1)
                                               (6,1,1)

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

A000212 counts the unimodal instead of coprime version.
A220377*6 is the strict case.
A307719 is the unordered version.
A337462 counts these compositions of any length.
A337563 counts the case of partitions with no 1's.
A337603 only requires the *distinct* parts to be pairwise coprime.
A337604 is the intersecting instead of coprime version.
A014612 ranks 3-part partitions.
A302696 ranks pairwise coprime partitions.
A327516 counts pairwise coprime partitions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A000217, A001399, A001840, A014311, A101268, A284825, A337562, A326675, A333227, A337601, A337602.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],CoprimeQ@@#&]],{n,0,30}]

A337561 Number of pairwise coprime strict compositions of n, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

1, 1, 0, 2, 2, 4, 8, 6, 16, 12, 22, 40, 40, 66, 48, 74, 74, 154, 210, 228, 242, 240, 286, 394, 806, 536, 840, 654, 1146, 1618, 2036, 2550, 2212, 2006, 2662, 4578, 4170, 7122, 4842, 6012, 6214, 11638, 13560, 16488, 14738, 15444, 16528, 25006, 41002, 32802

Offset: 0

Gus Wiseman, Sep 18 2020

nonn

			The a(1) = 1 through a(9) = 12 compositions (empty column shown as dot):
   (1)  .  (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

A072706 counts unimodal strict compositions.
A220377*6 counts these compositions of length 3.
A305713 is the unordered version.
A337462 is the not necessarily strict version.
A000740 counts relatively prime compositions, with strict case A332004.
A051424 counts pairwise coprime or singleton partitions.
A101268 considers all singletons to be coprime, with strict case A337562.
A178472 counts compositions with a common factor > 1.
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. A007359, A007360, A087087, A216652, A220377, A302569, A307719, A326675, A333227, A337461.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],#=={}||UnsameQ@@#&&CoprimeQ@@#&]],{n,0,10}]

a(n) = A337562(n) - 1 for n > 1.

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 }.

A337462 Number of pairwise coprime compositions of n, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

1, 1, 1, 3, 6, 12, 21, 37, 62, 100, 159, 253, 402, 634, 983, 1491, 2224, 3280, 4813, 7043, 10270, 14888, 21415, 30585, 43400, 61204, 85747, 119295, 164834, 226422, 309663, 422301, 574826, 781236, 1060181, 1436367, 1942588, 2622078, 3531151, 4742315, 6348410

Offset: 0

Gus Wiseman, Sep 18 2020

nonn

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

			The a(1) = 1 through a(5) = 12 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)  (1,3,1)
                                  (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..500

A000740 counts the relatively prime instead of pairwise coprime version.
A101268 considers all singletons to be coprime, with strict case A337562.
A327516 is the unordered version.
A333227 ranks these compositions, with complement A335239.
A337461 counts these compositions of length 3.
A337561 is the strict case.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A178472 counts compositions with a common factor.
A305713 counts strict pairwise coprime partitions.
A328673 counts pairwise non-coprime partitions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A337667 counts pairwise non-coprime compositions.
Cf. A001523, A007360, A087087, A220377, A302569, A307719, A326675, A335235, A335238, A337664.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],#=={}||CoprimeQ@@#&]],{n,0,10}]

For n > 1, a(n) = A101268(n) - 1.

A335235 Numbers k such that the k-th composition in standard order (A066099) is pairwise coprime, where a singleton is always considered coprime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 35, 37, 38, 39, 41, 44, 47, 48, 49, 50, 51, 52, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 71, 72, 75, 77, 78, 79, 80, 83, 89, 92, 95, 96, 97

Offset: 1

Gus Wiseman, May 28 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:
   1: (1)          20: (2,3)          48: (1,5)
   2: (2)          23: (2,1,1,1)      49: (1,4,1)
   3: (1,1)        24: (1,4)          50: (1,3,2)
   4: (3)          25: (1,3,1)        51: (1,3,1,1)
   5: (2,1)        27: (1,2,1,1)      52: (1,2,3)
   6: (1,2)        28: (1,1,3)        55: (1,2,1,1,1)
   7: (1,1,1)      29: (1,1,2,1)      56: (1,1,4)
   8: (4)          30: (1,1,1,2)      57: (1,1,3,1)
   9: (3,1)        31: (1,1,1,1,1)    59: (1,1,2,1,1)
  11: (2,1,1)      32: (6)            60: (1,1,1,3)
  12: (1,3)        33: (5,1)          61: (1,1,1,2,1)
  13: (1,2,1)      35: (4,1,1)        62: (1,1,1,1,2)
  14: (1,1,2)      37: (3,2,1)        63: (1,1,1,1,1,1)
  15: (1,1,1,1)    38: (3,1,2)        64: (7)
  16: (5)          39: (3,1,1,1)      65: (6,1)
  17: (4,1)        41: (2,3,1)        66: (5,2)
  18: (3,2)        44: (2,1,3)        67: (5,1,1)
  19: (3,1,1)      47: (2,1,1,1,1)    68: (4,3)

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

The version counting partitions is A051424, with strict case A007360.
The version for binary indices is A087087.
The version counting compositions is A101268.
The version for prime indices is A302569.
The case without singletons is A333227.
The complement is A335236.
Numbers whose binary indices are pairwise coprime are A326675.
Coprime partitions are counted by A327516.
All of the following pertain to compositions in standard order:
- Length is A000120.
- The parts are row k of A066099.
- Sum is A070939.
- Product is A124758.
- Reverse is A228351
- GCD is A326674.
- Heinz number is A333219.
- LCM is A333226.
Cf. A048793, A272919, A291166, A302569, A335236, A335237, A335238, A335239, A335240.

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

cp's OEIS Frontend

A228351 Triangle read by rows in which row n lists the compositions (ordered partitions) of n (see Comments lines for definition).

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A101268 Number of compositions of n into pairwise relatively prime parts.

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A327516 Number of integer partitions of n that are empty, (1), or have at least two parts and these parts are pairwise coprime.

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

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A333228 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are pairwise coprime, where a singleton is not considered coprime unless it is (1).

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

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A337561 Number of pairwise coprime strict compositions of n, where a singleton is not considered coprime unless it is (1).

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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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