A305713 -id:A305713 - cp's OEIS frontend

A337984 Heinz numbers of pairwise coprime integer partitions with no 1's, where a singleton is not considered coprime.

15, 33, 35, 51, 55, 69, 77, 85, 93, 95, 119, 123, 141, 143, 145, 155, 161, 165, 177, 187, 201, 205, 209, 215, 217, 219, 221, 249, 253, 255, 265, 287, 291, 295, 309, 323, 327, 329, 335, 341, 355, 381, 385, 391, 395, 403, 407, 411, 413, 415, 437, 447, 451, 465

Offset: 1

Gus Wiseman, Oct 22 2020

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
     15: {2,3}     155: {3,11}     265: {3,16}
     33: {2,5}     161: {4,9}      287: {4,13}
     35: {3,4}     165: {2,3,5}    291: {2,25}
     51: {2,7}     177: {2,17}     295: {3,17}
     55: {3,5}     187: {5,7}      309: {2,27}
     69: {2,9}     201: {2,19}     323: {7,8}
     77: {4,5}     205: {3,13}     327: {2,29}
     85: {3,7}     209: {5,8}      329: {4,15}
     93: {2,11}    215: {3,14}     335: {3,19}
     95: {3,8}     217: {4,11}     341: {5,11}
    119: {4,7}     219: {2,21}     355: {3,20}
    123: {2,13}    221: {6,7}      381: {2,31}
    141: {2,15}    249: {2,23}     385: {3,4,5}
    143: {5,6}     253: {5,9}      391: {7,9}
    145: {3,10}    255: {2,3,7}    395: {3,22}

A005117 is a superset.
A337485 counts these partitions.
A302568 considers singletons to be coprime.
A304711 allows 1's, with squarefree version A302797.
A337694 is the pairwise non-coprime instead of pairwise coprime version.
A007359 counts partitions into singleton or pairwise coprime parts with no 1's
A101268 counts pairwise coprime or singleton compositions, ranked by A335235.
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.
A337665 counts compositions whose distinct parts are pairwise coprime, ranked by A333228.
A337667 counts pairwise non-coprime compositions, ranked by A337666.
A337697 counts pairwise coprime compositions with no 1's.
A337983 counts pairwise non-coprime strict compositions, with unordered version A318717 ranked by A318719.
Cf. A051424, A056239, A087087, A112798, A200976, A220377, A302569, A303140, A303282, A328673, A328867.

Select[Range[1,100,2],SquareFreeQ[#]&&CoprimeQ@@PrimePi/@First/@FactorInteger[#]&]

Equals A302568\A000040.

A324750 Number of strict integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 3, 2, 4, 4, 4, 6, 8, 8, 11, 10, 15, 16, 19, 23, 27, 28, 35, 39, 47, 50, 63, 68, 77, 91, 102, 114, 130, 147, 169, 187, 213, 237, 268, 300, 336, 380, 422, 472, 525, 587, 647, 731, 810, 895, 996, 1102, 1227, 1355, 1498, 1661, 1818, 2020, 2221

Offset: 0

Gus Wiseman, Mar 15 2019

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 a(2) = 1 through a(17) = 15 strict integer partitions (A...H = 10...17):
  2  3  4  5  6   7   8   9   A   B    C    D    E    F    G    H
              42  43  62  54  64  65   75   76   86   87   97   98
                  52      63  73  83   84   85   95   96   A6   A7
                          72  82  542  93   94   A4   A5   C4   B6
                                       A2   A3   B3   B4   D3   C5
                                       642  B2   C2   C3   E2   D4
                                            643  752  D2   763  E3
                                            652  842  654  862  F2
                                                      762  943  854
                                                      843  A42  863
                                                      852       872
                                                                A43
                                                                A52
                                                                B42
                                                                6542

The subset version is A324739. The non-strict version is A324755. The Heinz number version is A324760. An infinite version is A324694.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A290822, A304360, A305713, A306844.
Cf. A324696, A324737, A324742, A324744, A324764.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,1]&&!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,0,30}]

A337697 Number of pairwise coprime compositions of n with no 1's, where a singleton is not considered coprime.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 4, 2, 4, 8, 8, 14, 10, 16, 12, 30, 38, 46, 46, 48, 52, 62, 152, 96, 156, 112, 190, 256, 338, 420, 394, 326, 402, 734, 622, 1150, 802, 946, 898, 1730, 1946, 2524, 2200, 2328, 2308, 3356, 5816, 4772, 5350, 4890, 6282, 6316, 12092, 8902

Offset: 0

Gus Wiseman, Oct 06 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. These compositions must be strict.

			The a(5) = 2 through a(12) = 14 compositions (empty column indicated by dot):
  (2,3)  .  (2,5)  (3,5)  (2,7)  (3,7)    (2,9)  (5,7)
  (3,2)     (3,4)  (5,3)  (4,5)  (7,3)    (3,8)  (7,5)
            (4,3)         (5,4)  (2,3,5)  (4,7)  (2,3,7)
            (5,2)         (7,2)  (2,5,3)  (5,6)  (2,7,3)
                                 (3,2,5)  (6,5)  (3,2,7)
                                 (3,5,2)  (7,4)  (3,4,5)
                                 (5,2,3)  (8,3)  (3,5,4)
                                 (5,3,2)  (9,2)  (3,7,2)
                                                 (4,3,5)
                                                 (4,5,3)
                                                 (5,3,4)
                                                 (5,4,3)
                                                 (7,2,3)
                                                 (7,3,2)

A022340 intersected with A333227 is a ranking sequence (using standard compositions A066099) for these compositions.
A212804 does not require coprimality, with unordered version A002865.
A337450 is the relatively prime instead of pairwise coprime version, with strict case A337451 and unordered version A302698.
A337462 allows 1's, with strict case A337561 (or A101268 with singletons), unordered version A327516 with Heinz numbers A302696, and 3-part case A337461.
A337485 is the unordered version (or A007359 with singletons considered coprime), with Heinz numbers A337984.
A337563 is the case of unordered triples.
Cf. A078374, A178472, A302568, A302697, A305713, A307719, A332004, A337562.

Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],!MemberQ[#,1]&&CoprimeQ@@#&]],{n,0,30}]

For n > 1, the version where singletons are considered coprime is a(n) + 1.

A343655 Number of pairwise coprime sets of divisors of n, where a singleton is not considered pairwise coprime unless it is {1}.

Original entry on oeis.org

1, 2, 2, 3, 2, 6, 2, 4, 3, 6, 2, 10, 2, 6, 6, 5, 2, 10, 2, 10, 6, 6, 2, 14, 3, 6, 4, 10, 2, 22, 2, 6, 6, 6, 6, 17, 2, 6, 6, 14, 2, 22, 2, 10, 10, 6, 2, 18, 3, 10, 6, 10, 2, 14, 6, 14, 6, 6, 2, 38, 2, 6, 10, 7, 6, 22, 2, 10, 6, 22, 2, 24, 2, 6, 10, 10, 6, 22, 2

Offset: 1

Gus Wiseman, Apr 26 2021

nonn

First differs from A015995 at a(210) = 88, A015995(210) = 86.

			For example, the a(n) subsets for n = 1, 2, 4, 6, 8, 12, 16, 24 are:
  {1}  {1}    {1}    {1}      {1}    {1}      {1}     {1}
       {1,2}  {1,2}  {1,2}    {1,2}  {1,2}    {1,2}   {1,2}
              {1,4}  {1,3}    {1,4}  {1,3}    {1,4}   {1,3}
                     {1,6}    {1,8}  {1,4}    {1,8}   {1,4}
                     {2,3}           {1,6}    {1,16}  {1,6}
                     {1,2,3}         {2,3}            {1,8}
                                     {3,4}            {2,3}
                                     {1,12}           {3,4}
                                     {1,2,3}          {3,8}
                                     {1,3,4}          {1,12}
                                                      {1,24}
                                                      {1,2,3}
                                                      {1,3,4}
                                                      {1,3,8}

The case of pairs is A063647.
The case of triples is A066620.
The version with empty sets and singletons is A225520.
A version for prime indices is A304711.
The version for strict integer partitions is A305713.
The version for subsets of {1..n} is A320426 = A276187 + 1.
The version for binary indices is A326675.
The version for integer partitions is A327516.
The version for standard compositions is A333227.
The maximal case is A343652.
The case without 1's is A343653.
The case without 1's with singletons is A343654.
The maximal case without 1's 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.
A325683 counts maximal Golomb rulers.
A326077 counts maximal pairwise indivisible sets.
Cf. A007360, A062319, A067824, A076078, A084422, A187106, A282935, A285572, A304709, A320423, A337485, A343659.

Table[Length[Select[Subsets[Divisors[n]],CoprimeQ@@#&]],{n,100}]

A066620 Number of unordered triples of distinct pairwise coprime divisors 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, 7, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 7, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 13, 0, 1, 2, 0, 1, 7, 0, 2, 1, 7, 0, 6, 0, 1, 2, 2, 1, 7, 0, 4, 0, 1, 0, 13, 1, 1, 1, 3, 0, 13, 1, 2, 1, 1, 1, 5, 0, 2, 2, 4, 0, 7, 0

Offset: 1

K. B. Subramaniam (kb_subramaniambalu(AT)yahoo.com) and Amarnath Murthy, Dec 24 2001

nonn

a(m) = a(n) if m and n have same factorization structure.

			a(24) = 3: the divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. The triples are (1, 2, 3), (1, 2, 9), (1, 3, 4).
a(30) = 7: the triples are (1, 2, 3), (1, 2, 5), (1, 3, 5), (2, 3, 5), (1, 3, 10), (1, 5, 6), (1, 2, 15).

Amarnath Murthy, Decomposition of the divisors of a natural number into pairwise coprime sets, Smarandache Notions Journal, vol. 12, No. 1-2-3, Spring 2001.pp 303-306.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Positions of zeros are A000961.
Positions of ones are A006881.
The version for subsets of {1..n} instead of divisors is A015617.
The non-strict ordered version is A048785.
The version for pairs of divisors is A063647.
The non-strict version (3-multisets) is A100565.
The version for partitions is A220377 (non-strict: A307719).
A version for sets of divisors of any size is A225520.
A000005 counts divisors.
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A007304 ranks 3-part strict partitions.
A014311 ranks 3-part compositions.
A014612 ranks 3-part partitions.
A018892 counts unordered pairs of coprime divisors (ordered: A048691).
A051026 counts pairwise indivisible subsets of {1..n}.
A337461 counts 3-part pairwise coprime compositions.
A338331 lists Heinz numbers of pairwise coprime partitions.
Cf. A007360, A023022, A084422, A276187, A282935, A305713, A337563, A337605, A343652, A343655.

Table[Length[Select[Subsets[Divisors[n],{3}],CoprimeQ@@#&]],{n,100}] (* Gus Wiseman, Apr 28 2021 *)

A066620(n) = (numdiv(n^3)-3*numdiv(n)+2)/6; \\ After Jovovic's formula. - Antti Karttunen, May 27 2017

from sympy import divisor_count as d
def a(n): return (d(n**3) - 3*d(n) + 2)/6 # Indranil Ghosh, May 27 2017

In the reference it is shown that if k is a squarefree number with r prime factors and m with (r+1) prime factors then a(m) = 4*a(k) + 2^k - 1.

a(n) = (tau(n^3)-3*tau(n)+2)/6. - Vladeta Jovovic, Nov 27 2004

More terms from Vladeta Jovovic, Apr 03 2003
Name corrected by Andrey Zabolotskiy, Dec 09 2020
Name corrected by Gus Wiseman, Apr 28 2021 (ordered version is 6*a(n))

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}]

A324749 Number of strict integer partitions of n containing no part > 1 whose prime indices all belong to the partition.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 4, 3, 4, 6, 6, 8, 11, 10, 14, 14, 19, 21, 26, 28, 35, 38, 44, 50, 60, 65, 79, 88, 98, 113, 131, 144, 165, 185, 211, 234, 268, 297, 334, 374, 420, 470, 525, 584, 649, 727, 801, 902, 998, 1100, 1220, 1357, 1500, 1657, 1833, 2029, 2220, 2462

Offset: 0

Gus Wiseman, Mar 15 2019

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 a(0) = 1 through a(10) = 6 strict integer partitions:
  ()  (1)  (2)  (3)  (4)    (5)  (6)    (7)    (8)    (9)    (10)
                     (3,1)       (4,2)  (4,3)  (6,2)  (5,4)  (6,4)
                                 (5,1)  (5,2)  (7,1)  (6,3)  (7,3)
                                        (6,1)         (7,2)  (8,2)
                                                             (9,1)
                                                             (6,3,1)

The subset version is A324738. The non-strict version is A324754. The Heinz number version is A324759. An infinite version is A324694.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A290822, A305713.
Cf. A324744, A324750, A324755, A324760.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,0,30}]

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[#]])&]

A343653 Number of non-singleton pairwise coprime nonempty sets of 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, 7, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 7, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 13, 0, 1, 2, 0, 1, 7, 0, 2, 1, 7, 0, 6, 0, 1, 2, 2, 1, 7, 0, 4, 0, 1, 0, 13, 1, 1

Offset: 1

Gus Wiseman, Apr 25 2021

nonn

First differs from A066620 at a(210) = 36, A066620(210) = 35.

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

The case of pairs is A089233.
The version with 1's, empty sets, and singletons is A225520.
The version for subsets of {1..n} is A320426.
The version for strict partitions is A337485.
The version for compositions is A337697.
The version for prime indices is A337984.
The maximal case with 1's is A343652.
The version with empty sets is a(n) + 1.
The version with singletons is A343654(n) - 1.
The version with empty sets and singletons is A343654.
The version with 1's is A343655.
The maximal case is A343660.
A018892 counts pairwise coprime unordered pairs of divisors.
A048691 counts pairwise coprime ordered pairs of divisors.
A048785 counts pairwise coprime ordered triples of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A100565 counts pairwise coprime unordered triples of divisors.
A305713 counts pairwise coprime non-singleton strict partitions.
A343659 counts maximal pairwise coprime subsets of {1..n}.
Cf. A007359, A067824, A074206, A076078, A084422, A187106, A285572, A324837, A326675, A327516, A338315.

Table[Length[Select[Subsets[Rest[Divisors[n]]],CoprimeQ@@#&]],{n,100}]

A337983 Number of compositions of n into distinct parts, any two of which have a common divisor > 1.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 3, 1, 3, 3, 5, 1, 13, 1, 13, 7, 19, 1, 35, 1, 59, 15, 65, 1, 117, 5, 133, 27, 195, 1, 411, 7, 435, 67, 617, 17, 941, 7, 1177, 135, 1571, 13, 2939, 31, 3299, 375, 4757, 13, 6709, 43, 8813, 643, 11307, 61, 16427, 123, 24331, 1203, 30461, 67

Offset: 0

Gus Wiseman, Oct 06 2020

nonn

Number of pairwise non-coprime strict compositions of n.

			The a(2) = 1 through a(15) = 7 compositions (A..F = 10..15):
  2  3  4  5  6   7  8   9   A   B  C    D  E    F
              24     26  36  28     2A      2C   3C
              42     62  63  46     39      4A   5A
                             64     48      68   69
                             82     84      86   96
                                    93      A4   A5
                                    A2      C2   C3
                                    246     248
                                    264     284
                                    426     428
                                    462     482
                                    624     824
                                    642     842

A318717 is the unordered version.
A318719 is the version for Heinz numbers of partitions.
A337561 is the pairwise coprime instead of pairwise non-coprime version, or A337562 if singletons are considered coprime.
A337605*6 counts these compositions of length 3.
A337667 is the non-strict version, ranked by A337666.
A337696 ranks these compositions.
A051185 and A305843 (covering) count pairwise intersecting set-systems.
A101268 counts pairwise coprime or singleton compositions.
A200976 and A328673 are the unordered version.
A233564 ranks strict compositions.
A318749 is the version for factorizations, with non-strict version A319786.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A335236 ranks compositions neither a singleton nor pairwise coprime.
A337462 counts pairwise coprime compositions.
A337694 lists numbers with no two relatively prime prime indices.
Cf. A082024, A284825, A302797, A305713, A319752, A327516, A333227, A335235, A336737, A337604.

stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&&stabQ[#,CoprimeQ]&]],{n,0,30}]

cp's OEIS Frontend

A337984 Heinz numbers of pairwise coprime integer partitions with no 1's, where a singleton is not considered coprime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A324750 Number of strict integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A337697 Number of pairwise coprime compositions of n with no 1's, where a singleton is not considered coprime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A343655 Number of pairwise coprime sets of divisors of n, where a singleton is not considered pairwise coprime unless it is {1}.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A066620 Number of unordered triples of distinct pairwise coprime divisors of n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A324749 Number of strict integer partitions of n containing no part > 1 whose prime indices all belong to the partition.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs