A302569 -id:A302569 - cp's OEIS frontend

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

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.

A038348 Expansion of (1/(1-x^2))*Product_{m>=0} 1/(1-x^(2m+1)).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 11, 14, 19, 24, 31, 39, 49, 61, 76, 93, 114, 139, 168, 203, 244, 292, 348, 414, 490, 579, 682, 801, 938, 1097, 1278, 1487, 1726, 1999, 2311, 2667, 3071, 3531, 4053, 4644, 5313, 6070, 6923, 7886, 8971, 10190, 11561

Offset: 0

N. J. A. Sloane

nonn

Number of partitions of n+2 with exactly one even part. - Vladeta Jovovic, Sep 10 2003
Also, number of partitions of n with at most one even part. - Vladeta Jovovic, Sep 10 2003
Also total number of parts, counted without multiplicity, in all partitions of n into odd parts, offset 1. - Vladeta Jovovic, Mar 27 2005
a(n) = Sum_{k>=1} k*A116674(n+1,k). - Emeric Deutsch, Feb 22 2006
Equals row sums of triangle A173305. - Gary W. Adamson, Feb 15 2010
Equals partial sums of A025147 (observed by Jonathan Vos Post, proved by several correspondents).
Conjecture: The n-th derivative of Gamma(x+1) at x = 0 has a(n+1) terms. For example, d^4/dx^4_(x = 0) Gamma(x+1) = 8*eulergamma*zeta(3) + eulergamma^4 + eulergamma^2*Pi^2 + 3*Pi^4/20 which has a(5) = 4 terms. - David Ulgenes, Dec 05 2023

			From _Gus Wiseman_, Sep 23 2019: (Start)
Also the number of integer partitions of n that are strict except possibly for any number of 1's. For example, the a(1) = 1 through a(7) = 11 partitions are:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (31)    (32)     (42)      (43)
             (111)  (211)   (41)     (51)      (52)
                    (1111)  (311)    (321)     (61)
                            (2111)   (411)     (421)
                            (11111)  (3111)    (511)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (31111)
                                               (211111)
                                               (1111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Cristina Ballantine and Mircea Merca, On identities of Watson type, Ars Mathematica Contemporanea (2019) Vol. 17, 277-290.
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
P. Flajolet and B. Salvy, Euler sums and contour integral representations, Experimental Mathematics, Vol. 7 Issue 1 (1998).
J. Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.), 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 2, with k=1. - _N. J. A. Sloane_, Aug 31 2014
Rebekah Ann Gilbert, A Fine Rediscovery, 2014.
Amrik Singh Nimbran and Paul Levrie, Series of the form Sum {a_n*binomial(2n, n)}, Math. Student (2023) Vol. 92, Nos. 3-4, 155-173. See pp. 10, 16.

Cf. A067588, A090867, A116674, A173305.

Cf. A000009, A007360, A051424, A259936, A266591, A302569, A306200.

f:=1/(1-x^2)/product(1-x^(2*j-1),j=1..32): fser:=series(f,x=0,62): seq(coeff(fser,x,n),n=0..58); # Emeric Deutsch, Feb 22 2006

mmax = 47; CoefficientList[ Series[ (1/(1-x^2))*Product[1/(1-x^(2m+1)), {m, 0, mmax}], {x, 0, mmax}], x] (* Jean-François Alcover, Jun 21 2011 *)

# uses[EulerTransform from A166861]
def g(n): return n % 2 if n > 2 else 1
a = EulerTransform(g)
print([a(n) for n in range(48)]) # Peter Luschny, Dec 04 2020

a(n) = A036469(n) - a(n-1) = Sum_{k=0..n} (-1)^k*A036469(n-k). - Vladeta Jovovic, Sep 10 2003
a(n) = A000009(n) + a(n-2). - Vladeta Jovovic, Feb 10 2004
G.f.: 1/((1-x^2)*Product_{j>=1} (1 - x^(2*j-1))). - Emeric Deutsch, Feb 22 2006
From Vaclav Kotesovec, Aug 16 2015: (Start)
a(n) ~ (1/2) * A036469(n).
a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). (End)
Euler transform of the sequence [1, 1, period(1, 0)] (A266591). - Georg Fischer, Dec 04 2020

A318717 Number of strict integer partitions of n in which no two parts are relatively prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 5, 1, 5, 4, 6, 1, 10, 1, 11, 6, 12, 1, 19, 3, 18, 8, 23, 1, 36, 2, 32, 13, 38, 7, 57, 2, 54, 19, 68, 3, 95, 3, 90, 33, 104, 3, 148, 7, 149, 40, 166, 5, 230, 17, 226, 56, 256, 6, 360, 9, 340, 84, 390, 25, 527, 11, 513, 109

Offset: 0

Gus Wiseman, Sep 02 2018

nonn

			The a(20) = 11 partitions:
  (20),
  (12,8), (14,6), (15,5), (16,4), (18,2),
  (10,6,4), (10,8,2), (12,6,2), (14,4,2),
  (8,6,4,2).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..600

Cf. A000009, A007359, A007360, A051185, A302569, A302797, A303140, A303280, A303283, A305713, A305843, A305854, A305713, A318715, A318719.

Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,And@@(GCD[##]>1&)@@@Select[Tuples[#,2],Less@@#&]]&]],{n,30}]

a(51)-a(69) from Alois P. Heinz, Sep 02 2018

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.

A259936 Number of ways to express the integer n as a product of its unitary divisors (A034444).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 5, 1, 2, 2, 1, 2, 5, 1, 2, 2, 5, 1, 2, 1, 2, 2, 2, 2, 5, 1, 2, 1, 2, 1, 5, 2, 2, 2, 2, 1, 5, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 5, 1, 2, 5

Offset: 1

Geoffrey Critzer, Jul 09 2015

nonn

Equivalently, a(n) is the number of ways to express the cyclic group Z_n as a direct sum of its Hall subgroups.  A Hall subgroup of a finite group G is a subgroup whose order is coprime to its index.
a(n) is the number of ways to partition the set of distinct prime factors of n.
Also the number of singleton or pairwise coprime factorizations of n. - Gus Wiseman, Sep 24 2019

			a(60) = 5 because we have: 60 = 4*3*5 = 4*15 = 3*20 = 5*12.
For n = 36, its unitary divisors are 1, 4, 9, 36. From these we obtain 36 either as 1*36 or 4*9, thus a(36) = 2. - _Antti Karttunen_, Oct 21 2017

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Wikipedia, Hall subgroup
Index entries for sequences computed from exponents in factorization of n

Cf. A000110, A001055, A001221, A034444, A089233, A258466, A281116, A285572.
Differs from A050320 for the first time at n=36.
Differs from A354870 for the first time at n=210, where a(210) = 15, while A354870(210) = 12.
Cf. A304716, A302569, A304711, A305079.
Related classes of factorizations:
- No conditions: A001055
- Strict: A045778
- Constant: A089723
- Distinct multiplicities: A255231
- Singleton or coprime: A259936
- Relatively prime: A281116
- Aperiodic: A303386
- Stable (indivisible): A305149
- Connected: A305193
- Strict relatively prime: A318721
- Uniform: A319269
- Intersecting: A319786
- Constant or distinct factors coprime: A327399
- Constant or relatively prime: A327400
- Coprime: A327517
- Not relatively prime: A327658
- Distinct factors coprime: A327695

map(combinat:-bell @ nops @ numtheory:-factorset, [$1..100]); # Robert Israel, Jul 09 2015

Table[BellB[PrimeNu[n]], {n, 1, 75}]
(* second program *)
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[#]==1||CoprimeQ@@#&]],{n,100}] (* Gus Wiseman, Sep 24 2019 *)

a(n) = my(t=omega(n), x='x, m=contfracpnqn(matrix(2, t\2, y, z, if( y==1, -z*x^2, 1 - (z+1)*x)))); polcoeff(1/(1 - x + m[2, 1]/m[1, 1]) + O(x^(t+1)), t) \\ Charles R Greathouse IV, Jun 30 2017

a(n) = A000110(A001221(n)).

a(n > 1) = A327517(n) + 1. - Gus Wiseman, Sep 24 2019

Incorrect comment removed by Antti Karttunen, Jun 11 2022

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[#]&]

A304709 Number of integer partitions of n whose distinct parts are pairwise coprime.

Original entry on oeis.org

1, 1, 2, 3, 6, 7, 13, 16, 23, 29, 42, 49, 69, 83, 102, 126, 161, 191, 239, 281, 336, 402, 484, 566, 672, 787, 919, 1067, 1251, 1449, 1684, 1934, 2223, 2554, 2920, 3341, 3821, 4344, 4928, 5586, 6334, 7163, 8091, 9100, 10228, 11492, 12902, 14449, 16167, 18058

Offset: 1

Gus Wiseman, May 17 2018

nonn

Two parts are coprime if they have no common divisor greater than 1. For partitions of length 1 note that (1) is coprime but (x) is not coprime for x > 1.

			The a(6) = 7 integer partitions of 6 whose distinct parts are pairwise coprime are (51), (411), (321), (3111), (2211), (21111), (111111).

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000005, A007359, A007360, A018783, A051424, A078374, A101268, A289508, A289509, A302569, A302696, A302698, A302796, A302797, A304711, A304712.

Table[Select[IntegerPartitions[n],CoprimeQ@@Union[#]&]//Length,{n,20}]

lista(nn)={local(Cache=Map());
  my(excl=vector(nn, n, sum(i=1, n-1, if(gcd(i,n)>1, 2^(n-i)))));
  my(c(n, m, b)=
     if(n==0, 1,
        while(m>n || bittest(b,0), m--; b>>=1);
        my(hk=[n, m, b], z);
        if(!mapisdefined(Cache, hk, &z),
          z = if(m, self()(n, m-1, b>>1) + self()(n-m, m, bitor(b, excl[m])), 0);
          mapput(Cache, hk, z)); z));
  my(a(n)=c(n, n, 0) + 1 - numdiv(n));
  for(n=1, nn, print1(a(n), ", "))
} \\ Andrew Howroyd, Nov 02 2019

a(n) = A304712(n) + 1 - A000005(n). - Andrew Howroyd, Nov 02 2019

A302697 Odd numbers whose prime indices are relatively prime. Heinz numbers of integer partitions with no 1's and with relatively prime parts.

Original entry on oeis.org

15, 33, 35, 45, 51, 55, 69, 75, 77, 85, 93, 95, 99, 105, 119, 123, 135, 141, 143, 145, 153, 155, 161, 165, 175, 177, 187, 195, 201, 205, 207, 209, 215, 217, 219, 221, 225, 231, 245, 249, 253, 255, 265, 275, 279, 285, 287, 291, 295, 297, 309, 315, 323, 327, 329

Offset: 1

Gus Wiseman, Apr 11 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of integer partitions with no 1's and with relatively prime parts begins:
015: (3,2)
033: (5,2)
035: (4,3)
045: (3,2,2)
051: (7,2)
055: (5,3)
069: (9,2)
075: (3,3,2)
077: (5,4)
085: (7,3)
093: (11,2)
095: (8,3)
099: (5,2,2)
105: (4,3,2)
119: (7,4)
123: (13,2)
135: (3,2,2,2)

Cf. A000837, A000961, A001222, A005117, A007359, A051424, A076078, A101268, A275024, A285572, A289509, A298748, A302568, A302569, A302696, A302698.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1,200,2],GCD@@primeMS[#]===1&]

A327076 Maximum divisor of n that is 1 or connected.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 9, 5, 11, 3, 13, 7, 5, 2, 17, 9, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 5, 31, 2, 11, 17, 7, 9, 37, 19, 39, 5, 41, 21, 43, 11, 9, 23, 47, 3, 49, 25, 17, 13, 53, 27, 11, 7, 57, 29, 59, 5, 61, 31, 63, 2, 65, 11, 67, 17, 23, 7, 71, 9, 73

Offset: 1

Gus Wiseman, Sep 05 2019

nonn

A number n with prime factorization n = prime(m_1)^s_1 * ... * prime(m_k)^s_k is connected if the simple labeled graph with vertex set {m_1,...,m_k} and edges between any two vertices with a common divisor greater than 1 is connected. Connected numbers are listed in A305078, which is the union of this sequence without 1.

Also the maximum MM-number (A302242) of a connected subset of the multiset of multisets with MM-number n.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

Positions of prime numbers are A302569.
Connected numbers are A305078.
Cf. A007947, A056239, A112798, A286518, A302242, A304716, A305079, A322338, A322390, A322391.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],GCD@@s[[#]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Max[Select[Divisors[n],Length[zsm[primeMS[#]]]<=1&]],{n,30}]

If n is in A305078, then a(n) = n.

A335238 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are not pairwise coprime, where a singleton is not coprime unless it is (1).

Original entry on oeis.org

0, 2, 4, 8, 10, 16, 32, 34, 36, 40, 42, 64, 69, 70, 81, 88, 98, 104, 128, 130, 136, 138, 139, 141, 142, 160, 162, 163, 168, 170, 177, 184, 197, 198, 209, 216, 226, 232, 256, 260, 261, 262, 274, 276, 277, 278, 279, 282, 283, 285, 286, 288, 290, 292, 296, 321

Offset: 1

Gus Wiseman, May 28 2020

nonn

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:
    0: ()          88: (2,1,4)      177: (2,1,4,1)
    2: (2)         98: (1,4,2)      184: (2,1,1,4)
    4: (3)        104: (1,2,4)      197: (1,4,2,1)
    8: (4)        128: (8)          198: (1,4,1,2)
   10: (2,2)      130: (6,2)        209: (1,2,4,1)
   16: (5)        136: (4,4)        216: (1,2,1,4)
   32: (6)        138: (4,2,2)      226: (1,1,4,2)
   34: (4,2)      139: (4,2,1,1)    232: (1,1,2,4)
   36: (3,3)      141: (4,1,2,1)    256: (9)
   40: (2,4)      142: (4,1,1,2)    260: (6,3)
   42: (2,2,2)    160: (2,6)        261: (6,2,1)
   64: (7)        162: (2,4,2)      262: (6,1,2)
   69: (4,2,1)    163: (2,4,1,1)    274: (4,3,2)
   70: (4,1,2)    168: (2,2,4)      276: (4,2,3)
   81: (2,4,1)    170: (2,2,2,2)    277: (4,2,2,1)

The complement is A333228.
Not ignoring repeated parts gives A335239.
Singleton or pairwise coprime partitions are counted by A051424.
Singleton or pairwise coprime sets are ranked by A087087.
Coprime partitions are counted by A327516.
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.
- Coprime compositions are A333227.
- Compositions whose distinct parts are coprime are A333228.
- Number of distinct parts is A334028.
Cf. A007360, A048793, A101268, A291166, A302569, A326675, A335235, A335236, A335237.

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

cp's OEIS Frontend

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

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A038348 Expansion of (1/(1-x^2))*Product_{m>=0} 1/(1-x^(2m+1)).

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A318717 Number of strict integer partitions of n in which no two parts are relatively prime.

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

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A259936 Number of ways to express the integer n as a product of its unitary divisors (A034444).

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

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A304709 Number of integer partitions of n whose distinct parts are pairwise coprime.

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