A302696 -id:A302696 - cp's OEIS frontend

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.

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[#]])&]

A366846 Numbers whose odd prime indices are relatively prime.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 55, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 85, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122

Offset: 1

Gus Wiseman, Oct 29 2023

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 odd prime indices of 115 are {3,9}, and these are not relatively prime, so 115 is not in the sequence.
The odd prime indices of 825 are {3,3,5}, and these are relatively prime, so 825 is in the sequence.

Including even indices gives A289509, ones of A289508, counted by A000837.
The complement when including even indices is A318978, counted by A018783.
The nonzero complement ranks the partitions counted by A366842.
The version for halved even indices is A366847.
The odd case is A366848.
The partitions with these Heinz numbers are counted by A366850.
A000041 counts integer partitions, strict A000009 (also into odds).
A112798 lists prime indices, length A001222, sum A056239.
A257992 counts even prime indices, odd A257991.
A366528 adds up odd prime indices, partition triangle A113685.
A366531 = 2*A366533 adds up even prime indices, triangle A113686/A174713.
Cf. A000720, A055396, A061395, A066208, A087436, A302696, A302697, A325698, A366843, A366844, A366849.

Select[Range[100], GCD@@Select[PrimePi/@First/@FactorInteger[#], OddQ]==1&]

A316469 Matula-Goebel numbers of unlabeled rooted identity RPMG-trees, meaning the Matula-Goebel numbers of the branches of any non-leaf node are relatively prime.

Original entry on oeis.org

1, 2, 6, 26, 78, 202, 606, 794, 2382, 2462, 2626, 7386, 7878, 8914, 10322, 12178, 26742, 30966, 32006, 36534, 42374, 43954, 47206, 80194, 96018, 115882, 127122, 131862, 141618, 149782, 158314, 160978, 184622, 217058, 240582, 248662, 260422, 347646, 449346

Offset: 1

Gus Wiseman, Jul 04 2018

nonn

A prime index of n is a number m such that prime(m) divides n. A number is in the sequence iff it is 1 or its prime indices are distinct, relatively prime, and already belong to the sequence.

			78 = prime(1)*prime(2)*prime(6) belongs to the sequence because the indices {1,2,6} are relatively prime, distinct, and already belong to the sequence.
The sequence of all identity RPMG-trees preceded by their Matula-Goebel numbers begins:
     1: o
     2: (o)
     6: (o(o))
    26: (o(o(o)))
    78: (o(o)(o(o)))
   202: (o(o(o(o))))
   606: (o(o)(o(o(o))))
   794: (o(o(o)(o(o))))
  2382: (o(o)(o(o)(o(o))))
  2462: (o(o(o(o(o)))))
  2626: (o(o(o))(o(o(o))))
  7386: (o(o)(o(o(o(o)))))
  7878: (o(o)(o(o))(o(o(o))))

Cf. A000081, A000837, A004111, A007097, A078374, A276625, A289509, A302696, A302796, A316467, A316470, A316471, A316474, A316494.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Or[#==1,And[SquareFreeQ[#],GCD@@primeMS[#]==1,And@@#0/@primeMS[#]]]&]

A327399 Number of factorizations of n that are constant or whose distinct factors are pairwise coprime.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 22 2019

nonn

First differs from A327400 at A327400(24) = 4, a(24) = 3.
From Jianing Song, Jun 09 2025: (Start)
Let n = (p_1)^(e_1) * ... * (p_r)^(e_r), then a(n) is the number of partitions of the multiset formed by e_1 1's, e_2 2's, ..., e_r r's such that each pair of parts is either equal or nonintersecting. Let's call such a partition a (e_1,...,e_r)-partition of {1,2,...,r}.
Note that every (e_1,...,e_r)-partition has a base partition by removing duplicates of parts and elements in each part (e.g., {{1,2,2},{1,2,2},{3,3},{4}} -> {{1,2},{3},{4}}), and the base partition is itself a partition on {1,2,...,r}. Since the number of partitions into identical parts of the multiset formed by e_{i_1} (i_1)'s, ..., e_{i_k} (i_k)'s is d(gcd(e_{i_1},...,e_{i_k})), where d = A000005, the number of (e_1,...,e_r)-partitions having base partition P of {1,2,...,r} is Product_{S in P} d(gcd_{i in S} (e_i)). As a result, the number (e_1,...,e_r)-partitions is Sum_{P is a partition of {1,2,...,r}} Product_{S in P} d(gcd_{i in S} (e_i)).
Examples:
  # of e_1-partitions = d(e_1);
  # of (e_1,e_2)-partitions = d(gcd(e_1,e_2)) + d(e_1)*d(e_2);
  # of (e_1,e_2,e_3)-partitions = d(gcd(e_1,e_2,e_3)) + d(gcd(e_1,e_2))*d(e_3) + d(gcd(e_1,e_3))*d(e_2) + d(gcd(e_2,e_3))*d(e_1) + d(e_1)*d(e_2)*d(e_3);
  # of (e_1,e_2,e_3,e_4)-partitions = d(gcd(e_1,e_2,e_3,e_4)) + (d(gcd(e_1,e_2,e_3))*d(e_4) + ...) + (d(gcd(e_1,e_2))*d(gcd(e_3,e_4)) + ...) + (d(gcd(e_1,e_2))*d(e_3)*d(e_4) + ...) + d(e_1)*d(e_2)*d(e_3)*d(e_4).
(End)

			The a(90) = 7 factorizations together with the corresponding multiset partitions of {1,2,2,3}:
  (2*3*3*5)  {{1},{2},{2},{3}}
  (2*5*9)    {{1},{3},{2,2}}
  (2*45)     {{1},{2,2,3}}
  (3*3*10)   {{2},{2},{1,3}}
  (5*18)     {{3},{1,2,2}}
  (9*10)     {{2,2},{1,3}}
  (90)       {{1,2,2,3}}

Gus Wiseman, Sequences counting and encoding certain classes of multisets

Constant factorizations are A089723.
Partitions whose distinct parts are pairwise coprime are A304709.
Factorizations that are constant or relatively prime are A327400.
See link for additional cross-references.
Cf. A007359, A050320, A051424, A281116, A302569, A302696, A304711.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Length[Union[#]]==1||CoprimeQ@@Union[#]&]],{n,100}]

a(n) = A327695(n) + A089723(n).

A337987 Odd numbers whose distinct prime indices are pairwise coprime, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

15, 33, 35, 45, 51, 55, 69, 75, 77, 85, 93, 95, 99, 119, 123, 135, 141, 143, 145, 153, 155, 161, 165, 175, 177, 187, 201, 205, 207, 209, 215, 217, 219, 221, 225, 245, 249, 253, 255, 265, 275, 279, 287, 291, 295, 297, 309, 323, 327, 329, 335, 341, 355, 363, 369

Offset: 1

Gus Wiseman, Oct 23 2020

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.

Also Heinz numbers of integer partitions with no 1's whose distinct parts are pairwise coprime (A338315). 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.

			The sequence of terms together with their prime indices begins:
     15: {2,3}      135: {2,2,2,3}    215: {3,14}
     33: {2,5}      141: {2,15}       217: {4,11}
     35: {3,4}      143: {5,6}        219: {2,21}
     45: {2,2,3}    145: {3,10}       221: {6,7}
     51: {2,7}      153: {2,2,7}      225: {2,2,3,3}
     55: {3,5}      155: {3,11}       245: {3,4,4}
     69: {2,9}      161: {4,9}        249: {2,23}
     75: {2,3,3}    165: {2,3,5}      253: {5,9}
     77: {4,5}      175: {3,3,4}      255: {2,3,7}
     85: {3,7}      177: {2,17}       265: {3,16}
     93: {2,11}     187: {5,7}        275: {3,3,5}
     95: {3,8}      201: {2,19}       279: {2,2,11}
     99: {2,2,5}    205: {3,13}       287: {4,13}
    119: {4,7}      207: {2,2,9}      291: {2,25}
    123: {2,13}     209: {5,8}        295: {3,17}

A304711 is the not necessarily odd version, with squarefree case A302797.
A337694 is a pairwise non-coprime instead of pairwise coprime version.
A337984 is the squarefree case.
A338315 counts the partitions with these Heinz numbers.
A338316 considers singletons coprime.
A007359 counts partitions into singleton or pairwise coprime parts with no 1's, with Heinz numbers A302568.
A304709 counts partitions whose distinct parts are pairwise coprime.
A327516 counts pairwise coprime partitions, with Heinz numbers 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.
A318717 counts pairwise non-coprime strict partitions, with Heinz numbers A318719.
Cf. A005408, A051424, A056239, A087087, A112798, A200976, A289508, A289509, A302569, A303282, A328867, A337485.

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

A338315 Number of integer partitions of n with no 1's whose distinct parts are pairwise coprime, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 3, 2, 4, 4, 10, 6, 15, 13, 16, 21, 31, 29, 43, 41, 50, 63, 79, 81, 99, 113, 129, 145, 179, 197, 228, 249, 284, 328, 363, 418, 472, 522, 581, 655, 741, 828, 921, 1008, 1123, 1259, 1407, 1546, 1709, 1889, 2077, 2292, 2554, 2799, 3061, 3369

Offset: 0

Gus Wiseman, Oct 23 2020

nonn

The Heinz numbers of these partitions are given by A337987. 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.

			The a(5) = 1 through a(13) = 15 partitions (empty column indicated by dot, A = 10, B = 11):
  32   .  43    53    54     73     65      75      76
          52    332   72     433    74      543     85
          322         522    532    83      552     94
                      3222   3322   92      732     A3
                                    443     5322    B2
                                    533     33222   544
                                    722             553
                                    3332            733
                                    5222            922
                                    32222           4333
                                                    5332
                                                    7222
                                                    33322
                                                    52222
                                                    322222

A200976 is a pairwise non-coprime instead of pairwise coprime version.
A304709 allows 1's, with strict case A305713 and Heinz numbers A304711.
A318717 counts pairwise non-coprime strict partitions.
A337485 is the strict version, with Heinz numbers A337984.
A337987 gives the Heinz numbers of these partitions.
A338317 considers singletons coprime, with Heinz numbers A338316.
A007359 counts singleton or pairwise coprime partitions with no 1's.
A327516 counts pairwise coprime partitions, ranked by A302696.
A328673 counts partitions with no two distinct parts relatively prime.
A337462 counts pairwise coprime compositions, ranked by A333227.
A337561 counts pairwise coprime strict compositions.
A337665 counts compositions whose distinct parts are pairwise coprime.
A337667 counts pairwise non-coprime compositions, ranked by A337666.
A337697 counts pairwise coprime compositions with no 1's.
Cf. A051424, A101268, A220377, A302568, A328867, A333228, A337563.

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

A355738 Least k such that there are exactly n ways to choose a sequence of divisors, one of each prime index of k (with multiplicity), such that the result has no common divisor > 1.

Original entry on oeis.org

1, 2, 6, 9, 15, 49, 35, 27, 45, 98, 63, 105, 171, 117, 81, 135, 245, 343, 273, 549, 189, 1083, 315, 5618, 741, 686, 507, 513, 351, 243, 405, 7467, 6419, 5575, 735, 6859, 1813, 3231, 1183, 1197, 3537, 819, 1647, 567, 945, 2197, 8397, 3211, 1715, 3249, 3367

Offset: 1

Gus Wiseman, Jul 21 2022

nonn

This is the position of first appearance of n in A355737.

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 terms together with their prime indices begin:
     1: {}
     2: {1}
     6: {1,2}
     9: {2,2}
    15: {2,3}
    49: {4,4}
    35: {3,4}
    27: {2,2,2}
    45: {2,2,3}
    98: {1,4,4}
    63: {2,2,4}
   105: {2,3,4}
   171: {2,2,8}
   117: {2,2,6}
    81: {2,2,2,2}
   135: {2,2,2,3}
For example, the choices for a(12) = 105 are:
  (1,1,1)  (1,3,2)  (2,1,4)
  (1,1,2)  (1,3,4)  (2,3,1)
  (1,1,4)  (2,1,1)  (2,3,2)
  (1,3,1)  (2,1,2)  (2,3,4)

Wikipedia, Coprime integers.

Not requiring coprimality gives A355732, firsts of A355731.
Positions of first appearances in A355737.
A000005 counts divisors.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A120383 lists numbers divisible by all of their prime indices.
A289508 gives GCD of prime indices.
A289509 ranks relatively prime partitions, odd A302697, squarefree A302796.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A007359, A051424, A076610, A302696, A302698, A355733, A355735, A355739, A355741, A355748.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
az=Table[Length[Select[Tuples[Divisors/@primeMS[n]],GCD@@#==1&]],{n,100}];
Table[Position[az+1,k][[1,1]],{k,mnrm[az+1]}]

A366847 Numbers whose halved even prime indices are nonempty and relatively prime.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 91, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174

Offset: 1

Gus Wiseman, Oct 31 2023

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.

Consists of powers of 2 times elements of the odd restriction A366849.

			The even prime indices of 91 are {4,6}, halved {2,3}, which are relatively prime, so 91 is in the sequence.
The prime indices of 665 are {3,4,8}, even {4,8}, halved {2,4}, which are not relatively prime, so 665 is not in the sequence.
The terms together with their prime indices begin:
    3: {2}
    6: {1,2}
    9: {2,2}
   12: {1,1,2}
   15: {2,3}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   33: {2,5}
   36: {1,1,2,2}
   39: {2,6}
   42: {1,2,4}
   45: {2,2,3}
   48: {1,1,1,1,2}

Including odd indices gives A289509, ones of A289508, counted by A000837.
The complement including odd indices is A318978, counted by A018783.
The partitions with these ranks are counted by A366845.
A version for odd indices A366846, counted by A366850.
The odd restriction is A366849.
A000041 counts integer partitions, strict A000009 (also into odds).
A035363 counts partitions into all even parts, ranks A066207.
A112798 lists prime indices, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A257992 counts even prime indices, odd A257991.
A366528 adds up odd prime indices, partition triangle A113685.
A366531 = 2*A366533 adds up even prime indices, triangle A113686/A174713.
Cf. A000720, A055396, A061395, A066208, A168532, A302696, A302697, A325698, A366842, A366843, A366844, A366848.

Select[Range[100],GCD@@Select[PrimePi/@First/@FactorInteger[#],EvenQ]/2==1&]

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.

A327512 Maximum divisor of n that is 1, 2, or a nonprime number whose prime indices are pairwise coprime.

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 1, 8, 1, 10, 1, 12, 1, 14, 15, 16, 1, 6, 1, 20, 1, 22, 1, 24, 1, 26, 1, 28, 1, 30, 1, 32, 33, 34, 35, 12, 1, 38, 1, 40, 1, 14, 1, 44, 15, 46, 1, 48, 1, 10, 51, 52, 1, 6, 55, 56, 1, 58, 1, 60, 1, 62, 1, 64, 1, 66, 1, 68, 69, 70, 1, 24, 1, 74

Offset: 1

Gus Wiseman, Sep 19 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. Numbers that are 1, 2, or a nonprime number whose prime indices are pairwise coprime are listed in A302696, which is the union of this sequence.

a(n) is the greatest term of A302696 which divides n. - Antti Karttunen, Dec 06 2021

			The divisors of 72 that are 1, 2, or nonprime numbers whose prime indices are pairwise coprime are: {1, 2, 4, 6, 8, 12, 24}, so a(72) = 24.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Gus Wiseman, Sequences counting and encoding certain classes of multisets
Index entries for sequences computed from indices in prime factorization

See link for additional cross-references.

Cf. A000005, A006530, A056239, A112798, A289509, A302569, A302696, A304711.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Max[Select[Divisors[n],#==1||CoprimeQ@@primeMS[#]&]],{n,100}]

isA302696(n) = if(isprimepower(n),!(n%2), if(!issquarefree(n>>valuation(n,2)), 0, my(pis=apply(primepi,factor(n)[,1])); (lcm(pis)==factorback(pis))));
A327512(n) = vecmax(select(isA302696,divisors(n))); \\ Antti Karttunen, Dec 06 2021

cp's OEIS Frontend

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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A366846 Numbers whose odd prime indices are relatively prime.

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A316469 Matula-Goebel numbers of unlabeled rooted identity RPMG-trees, meaning the Matula-Goebel numbers of the branches of any non-leaf node are relatively prime.

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A327399 Number of factorizations of n that are constant or whose distinct factors are pairwise coprime.

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A337987 Odd numbers whose distinct prime indices are pairwise coprime, where a singleton is not considered coprime unless it is (1).

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A338315 Number of integer partitions of n with no 1's whose distinct parts are pairwise coprime, where a singleton is not considered coprime unless it is (1).

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A355738 Least k such that there are exactly n ways to choose a sequence of divisors, one of each prime index of k (with multiplicity), such that the result has no common divisor > 1.

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A366847 Numbers whose halved even prime indices are nonempty and relatively prime.

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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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