A305148 -id:A305148 - cp's OEIS frontend

A051026 Number of primitive subsequences of {1, 2, ..., n}.

1, 2, 3, 5, 7, 13, 17, 33, 45, 73, 103, 205, 253, 505, 733, 1133, 1529, 3057, 3897, 7793, 10241, 16513, 24593, 49185, 59265, 109297, 163369, 262489, 355729, 711457, 879937, 1759873, 2360641, 3908545, 5858113, 10534337, 12701537, 25403073, 38090337, 63299265, 81044097, 162088193, 205482593, 410965185, 570487233, 855676353

Offset: 0

Eric W. Weisstein

a(n) counts all subsequences of {1, ..., n} in which no term divides any other. If n is a prime a(n) = 2*a(n-1)-1 because for each subsequence s counted by a(n-1) two different subsequences are counted by a(n): s and s,n. There is only one exception: 1,n is not a primitive subsequence because 1 divides n. For all n>1: a(n) < 2*a(n-1). - Alois P. Heinz, Mar 07 2011

Maximal primitive subsets are counted by A326077. - Gus Wiseman, Jun 07 2019

			a(4) = 7, the primitive subsequences (including the empty sequence) are: (), (1), (2), (3), (4), (2,3), (3,4).
a(5) = 13 = 2*7-1, the primitive subsequences are: (), (5), (1), (2), (2,5), (3), (3,5), (4), (4,5), (2,3), (2,3,5), (3,4), (3,4,5).
From _Gus Wiseman_, Jun 07 2019: (Start)
The a(0) = 1 through a(5) = 13 primitive (pairwise indivisible) subsets:
  {}  {}   {}   {}     {}     {}
      {1}  {1}  {1}    {1}    {1}
           {2}  {2}    {2}    {2}
                {3}    {3}    {3}
                {2,3}  {4}    {4}
                       {2,3}  {5}
                       {3,4}  {2,3}
                              {2,5}
                              {3,4}
                              {3,5}
                              {4,5}
                              {2,3,5}
                              {3,4,5}
a(n) is also the number of subsets of {1..n} containing all of their pairwise products <= n as well as any quotients of divisible elements. For example, the a(0) = 1 through a(5) = 13 subsets are:
  {}  {}   {}     {}       {}         {}
      {1}  {1}    {1}      {1}        {1}
           {1,2}  {1,2}    {1,3}      {1,3}
                  {1,3}    {1,4}      {1,4}
                  {1,2,3}  {1,2,4}    {1,5}
                           {1,3,4}    {1,2,4}
                           {1,2,3,4}  {1,3,4}
                                      {1,3,5}
                                      {1,4,5}
                                      {1,2,3,4}
                                      {1,2,4,5}
                                      {1,3,4,5}
                                      {1,2,3,4,5}
Also the number of subsets of {1..n} containing all of their multiples <= n. For example, the a(0) = 1 through a(5) = 13 subsets are:
  {}  {}   {}     {}       {}         {}
      {1}  {2}    {2}      {3}        {3}
           {1,2}  {3}      {4}        {4}
                  {2,3}    {2,4}      {5}
                  {1,2,3}  {3,4}      {2,4}
                           {2,3,4}    {3,4}
                           {1,2,3,4}  {3,5}
                                      {4,5}
                                      {2,3,4}
                                      {2,4,5}
                                      {3,4,5}
                                      {2,3,4,5}
                                      {1,2,3,4,5}
(End)
From _Gus Wiseman_, Mar 12 2024: (Start)
Also the number of subsets of {1..n} containing all divisors of the elements. For example, the a(0) = 1 through a(6) = 17 subsets are:
  {}  {}   {}     {}       {}         {}
      {1}  {1}    {1}      {1}        {1}
           {1,2}  {1,2}    {1,2}      {1,2}
                  {1,3}    {1,3}      {1,3}
                  {1,2,3}  {1,2,3}    {1,5}
                           {1,2,4}    {1,2,3}
                           {1,2,3,4}  {1,2,4}
                                      {1,2,5}
                                      {1,3,5}
                                      {1,2,3,4}
                                      {1,2,3,5}
                                      {1,2,4,5}
                                      {1,2,3,4,5}
(End)

Blanchet-Sadri, Francine. Algorithmic combinatorics on partial words. Chapman & Hall/CRC, Boca Raton, FL, 2008. ii+385 pp. ISBN: 978-1-4200-6092-8; 1-4200-6092-9 MR2384993 (2009f:68142). See p. 320. - N. J. A. Sloane, Apr 06 2012

Juliana Couras, Ricardo Jesus, and Tomás Oliveira e Silva, Table of n, a(n) for n = 0..800 (terms up to n=80 from Alois P. Heinz)
Marcel K. Goh and Jonah Saks, Alternating-sum statistics for certain sets of integers, arXiv:2206.12535 [math.CO], 2022.
Nathan McNew, Counting primitive subsets and other statistics of the divisor graph of {1,2,..,n}, arXiv:1808.04923 [math.NT], 2018.
Richárd Palincza, Counting type and extremal problems from Arithmetic Combinatorics, Ph. D. Thesis, Budapest Univ. Tech. Econ. (Hungary, 2024).
Eric Weisstein's World of Mathematics, Primitive Sequence

Cf. A007865, A054519, A096827, A103580, A303362, A305148.
Cf. A326023, A326076, A326077, A326081, A326082, A326083, A326117.
Cf. A037031, A051026, A355740, A368110, A370585.

with(numtheory):
b:= proc(s) option remember; local n;
      n:= max(s[]);
      `if`(n<0, 1, b(s minus {n}) + b(s minus divisors(n)))
    end:
bb:= n-> b({$2..n} minus divisors(n)):
sb:= proc(n) option remember; `if`(n<2, 0, bb(n) + sb(n-1)) end:
a:= n-> `if`(n=0, 1, `if`(isprime(n), 2*a(n-1)-1, 2+sb(n))):
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 07 2011

b[s_] := b[s] = With[{n=Max[s]}, If[n < 0, 1, b[Complement[s, {n}]] + b[Complement[s, Divisors[n]]]]];
bb[n_] := b[Complement[Range[2, n], Divisors[n]]];
sb[n_] := sb[n] = If[n < 2, 0, bb[n] + sb[n-1]];
a[n_] := If[n == 0, 1, If[PrimeQ[n], 2a[n-1] - 1, 2 + sb[n]]]; Table[a[n], {n, 0, 37}]
(* Jean-François Alcover, Jul 27 2011, converted from Maple *)
Table[Length[Select[Subsets[Range[n]], SubsetQ[#,Select[Union@@Table[#*i,{i,n}],#<=n&]]&]],{n,10}] (* Gus Wiseman, Jun 07 2019 *)
Table[Length[Select[Subsets[Range[n]], #==Union@@Divisors/@#&]],{n,0,10}] (* Gus Wiseman, Mar 12 2024 *)

More terms from David Wasserman, May 02 2002

a(32)-a(37) from Donovan Johnson, Aug 11 2010

A323092 Number of double-free integer partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 10, 14, 17, 24, 30, 40, 50, 66, 81, 104, 128, 161, 197, 246, 300, 369, 446, 546, 656, 796, 952, 1148, 1366, 1637, 1940, 2311, 2730, 3234, 3806, 4489, 5262, 6181, 7225, 8454, 9846, 11484, 13335, 15499, 17948, 20796, 24017, 27751, 31970, 36837

Offset: 0

Gus Wiseman, Jan 04 2019

nonn

An integer partition is double-free if no part is twice any other part.

			The a(1) = 1 through a(8) = 14 double-free integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (31)    (41)     (51)      (52)       (53)
                    (1111)  (311)    (222)     (61)       (62)
                            (11111)  (411)     (322)      (71)
                                     (3111)    (331)      (332)
                                     (111111)  (511)      (431)
                                               (4111)     (611)
                                               (31111)    (2222)
                                               (1111111)  (3311)
                                                          (5111)
                                                          (41111)
                                                          (311111)
                                                          (11111111)

Alois P. Heinz, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Double-Free Set

Cf. A018819, A051424, A101417, A120641, A276431, A305148, A323093, A323094.

Table[Length[Select[IntegerPartitions[n],Intersection[#,2*#]=={}&]],{n,30}]

A320340 Heinz numbers of double-free integer partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Jan 07 2019

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
An integer partition is double-free if no part is twice any other part.
Also numbers n such that if prime(m) divides n then prime(2m) does not divide n, i.e., numbers not divisible by any element of A319613.

			The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (), (1), (2), (11), (3), (4), (111), (22), (31), (5), (6), (41), (32), (1111), (7), (8), (311), (51), (9), (33), (61), (222), (411).

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Double-Free Set

Cf. A018819, A056239, A087897, A101417, A112798, A120641, A276431, A305148, A319613, A323092, A323093.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[primeMS[#],2*primeMS[#]]=={}&]

A328673 Number of integer partitions of n in which no two distinct parts are relatively prime.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 2, 6, 4, 9, 2, 15, 2, 17, 10, 23, 2, 39, 2, 46, 18, 58, 2, 95, 8, 103, 31, 139, 2, 219, 3, 232, 59, 299, 22, 452, 4, 492, 104, 645, 5, 920, 5, 1006, 204, 1258, 8, 1785, 21, 1994, 302, 2442, 11, 3366, 71, 3738, 497, 4570, 18, 6253, 24, 6849

Offset: 0

Gus Wiseman, Oct 29 2019

nonn

A partition with no two distinct parts relatively prime is said to be intersecting.

			The a(1) = 1 through a(10) = 9 partitions (A = 10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        63         55
              1111         42               62        333        64
                           222              422       111111111  82
                           111111           2222                 442
                                            11111111             622
                                                                 4222
                                                                 22222
                                                                 1111111111

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..350

The Heinz numbers of these partitions are A328867 (strict case is A318719).
The relatively prime case is A328672.
The strict case is A318717.
The version for non-isomorphic multiset partitions is A319752.
The version for set-systems is A305843.
The version involving all parts (not just distinct ones) is A200976.
Cf. A000837, A202425, A305148, A305854, A306006, A316476, A326910.

Table[Length[Select[IntegerPartitions[n],And@@(GCD[##]>1&)@@@Subsets[Union[#],{2}]&]],{n,0,20}]

a(n > 0) = A200976(n) + 1.

A319564 Number of T_0 integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 21, 29, 40, 53, 73, 95, 128, 168, 221, 282, 368, 466, 599, 759, 962, 1201, 1513, 1881, 2345, 2901, 3590, 4407, 5416, 6614, 8083, 9827, 11937, 14442, 17458, 21021, 25299, 30347, 36363, 43438, 51843, 61705, 73384, 87054, 103149, 121949

Offset: 0

Gus Wiseman, Sep 23 2018

nonn

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. For an integer partition the T_0 condition means the dual of the multiset partition obtained by factoring each part into prime numbers is strict (no repeated blocks).

Also the number of integer partitions of n with no equivalent primes. In an integer partition, two primes are equivalent if each part has in its prime factorization the same multiplicity of both primes. For example, in (6,5) the primes {2,3} are equivalent. See A316978 for more examples.

Cf. A000009, A000041, A001970, A007716, A059201, A305148, A316978, A316979, A316983, A319558, A319616, A319728.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}]
Table[Length[Select[IntegerPartitions[n],UnsameQ@@dual[primeMS/@#]&]],{n,20}]

A200976 Number of partitions of n such that each pair of parts (if any) has a common factor.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 4, 1, 5, 3, 8, 1, 14, 1, 16, 9, 22, 1, 38, 1, 45, 17, 57, 1, 94, 7, 102, 30, 138, 1, 218, 2, 231, 58, 298, 21, 451, 3, 491, 103, 644, 4, 919, 4, 1005, 203, 1257, 7, 1784, 20, 1993, 301, 2441, 10, 3365, 70, 3737, 496, 4569, 17, 6252, 23, 6848

Offset: 0

Alois P. Heinz, Nov 29 2011

nonn

a(n) is different from A018783(n) for n = 0, 31, 37, 41, 43, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, ... .

Every pair of (possibly equal) parts has a common factor > 1. These partitions are said to be (pairwise) intersecting. - Gus Wiseman, Nov 04 2019

			a(0) = 1: [];
a(4) = 2: [2,2], [4];
a(9) = 3: [3,3,3], [3,6], [9];
a(31) = 2: [6,10,15], [31];
a(41) = 4: [6,10,10,15], [6,15,20], [6,14,21], [41].

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..350 (terms 0..250 from Alois P. Heinz)
L. Naughton, G. Pfeiffer, Integer Sequences Realized by the Subgroup Pattern of the Symmetric Group, J. Int. Seq. 16 (2013) #13.5.8

Cf. A018783.
The version with only distinct parts compared is A328673.
The relatively prime case is A202425.
The strict case is A318717.
The version for non-isomorphic multiset partitions is A319752.
The version for set-systems is A305843.
Cf. A000837, A305148, A305854, A306006, A316476, A328672, A328867, A328868.

b:= proc(n, j, s) local ok, i;
      if n=0 then 1
    elif j<2 then 0
    else ok:= true;
         for i in s while ok do ok:= evalb(igcd(i, j)<>1) od;
         `if`(ok, add(b(n-j*k, j-1, [s[], j]), k=1..n/j), 0) +b(n, j-1, s)
      fi
    end:
a:= n-> b(n, n, []):
seq(a(n), n=0..62);

b[n_, j_, s_] := Module[{ok, i, is}, Which[n == 0, 1, j < 2, 0, True, ok = True; For[is = 1, is <= Length[s] && ok, is++, i = s[[is]]; ok = GCD[i, j] != 1]; If[ok, Sum[b[n-j*k, j-1, Append[s, j]], {k, 1, n/j}], 0] + b[n, j-1, s]]]; a[n_] := b[n, n, {}]; Table[a[n], {n, 0, 62}] (* Jean-François Alcover, Dec 26 2013, translated from Maple *)
Table[Length[Select[IntegerPartitions[n],And[And@@(GCD[##]>1&)@@@Select[Tuples[Union[#],2],LessEqual@@#&]]&]],{n,0,20}] (* Gus Wiseman, Nov 04 2019 *)

a(n > 0) = A328673(n) - 1. - Gus Wiseman, Nov 04 2019

A305149 Number of factorizations of n whose distinct factors are pairwise indivisible and greater than 1.

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

Offset: 1

Gus Wiseman, May 26 2018

nonn

			The a(60) = 8 factorizations are (2*2*3*5), (2*2*15), (3*4*5), (3*20), (4*15), (5*12), (6*10), (60). Missing from this list are (2*3*10), (2*5*6), (2*30).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A001055, A001970, A007716, A034444, A045778, A259936, A281116, A285572, A302242, A303386, A303431, A305001, A305148, A305150.

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],Select[Tuples[Union[#],2],UnsameQ@@#&&Divisible@@#&]=={}&]],{n,100}]

pairwise_indivisible(v) = { for(i=1,#v,for(j=i+1,#v,if(!(v[j]%v[i]),return(0)))); (1); };
A305149(n, m=n, facs=List([])) = if(1==n, pairwise_indivisible(Set(facs)), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A305149(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Oct 08 2018

More terms from Antti Karttunen, Oct 08 2018

A371128 Number of strict integer partitions of n containing all distinct divisors of all parts.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 1, 2, 1, 2, 2, 3, 3, 3, 5, 3, 5, 6, 7, 7, 8, 8, 9, 12, 13, 13, 14, 15, 16, 19, 23, 25, 26, 26, 27, 36, 37, 40, 42, 46, 50, 55, 66, 65, 71, 71, 82, 90, 102, 103, 114, 117, 130, 147, 154, 166, 176, 182, 194, 228, 239, 259, 267, 287, 307, 336

Offset: 0

Gus Wiseman, Mar 18 2024

nonn

Also strict integer partitions such that the number of parts is equal to the number of distinct divisors of all parts.

			The a(9) = 1 through a(19) = 7 partitions (A..H = 10..17):
  531  721   731   B1    751   D1    B31    D21    B51    H1     B71
       4321  5321  5421  931   B21   7521   7531   D31    9531   D51
                   6321  7321  7421  8421   64321  B321   A521   B521
                                     9321          65321  B421   D321
                                     54321         74321  75321  75421
                                                          84321  76321
                                                                 94321

The LHS is represented by A001221, distinct case of A001222.
The RHS is represented by A370820, for prime factors A303975.
Strict case of A371130 (ranks A370802) and A371178 (ranks A371177).
The complement is counted by A371180, non-strict A371132.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
A305148 counts partitions without divisors, strict A303362, ranks A316476.
Cf. A000837, A003963, A239312, A285573, A319055, A370803, A370808, A371171, A371172, A371173.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&SubsetQ[#,Union@@Divisors/@#]&]],{n,0,30}]

A371177 Positive integers whose prime indices include all distinct divisors of all prime indices.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 30, 32, 34, 36, 40, 42, 44, 48, 50, 54, 60, 62, 64, 66, 68, 72, 80, 82, 84, 88, 90, 96, 100, 102, 108, 110, 118, 120, 124, 126, 128, 132, 134, 136, 144, 150, 160, 162, 164, 166, 168, 170, 176, 180, 186, 192, 198, 200

Offset: 1

Gus Wiseman, Mar 18 2024

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 positive integers with as many distinct prime factors (A001221) as distinct divisors of prime indices (A370820).

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   34: {1,7}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   48: {1,1,1,1,2}

The LHS is A001221, distinct case of A001222.
The RHS is A370820, for prime factors A303975.
For bigomega on the LHS we have A370802, counted by A371130.
For divisors on the LHS we have A371165, counted by A371172.
Partitions of this type are counted by A371178, strict A371128.
The complement is A371179, counted by A371132.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
A305148 counts partitions without divisors, strict A303362, ranks A316476.
Cf. A000837, A003963, A239312, A285573, A355529, A370813, A371168.

Select[Range[100],PrimeNu[#]==Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

A001221(a(n)) = A370820(a(n)).

A087086 Primitive sets of integers, each subset mapped onto a unique binary integer, values here shown in decimal.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 16, 18, 20, 22, 24, 28, 32, 40, 48, 56, 64, 66, 68, 70, 72, 76, 80, 82, 84, 86, 88, 92, 96, 104, 112, 120, 128, 132, 144, 148, 160, 176, 192, 196, 208, 212, 224, 240, 256, 258, 264, 272, 274, 280, 288, 296, 304, 312, 320, 322, 328, 336, 338, 344

Offset: 0

Alan Sutcliffe (alansut(AT)ntlworld.com), Aug 14 2003

A primitive set of integers has no pair of elements one of which divides the other. Each element i in a subset contributes 2^(i-1) to the binary value for that subset. The integers missing from the sequence correspond to nonprimitive subsets.

			a(10)=22 since the 10th primitive set counting from 0 is {5,3,2}, which maps onto 10110 binary = 22 decimal.
From _Gus Wiseman_, Oct 31 2019: (Start)
The sequence of terms together with their binary expansions and binary indices begins:
   0:       0 ~ {}
   1:       1 ~ {1}
   2:      10 ~ {2}
   4:     100 ~ {3}
   6:     110 ~ {2,3}
   8:    1000 ~ {4}
  12:    1100 ~ {3,4}
  16:   10000 ~ {5}
  18:   10010 ~ {2,5}
  20:   10100 ~ {3,5}
  22:   10110 ~ {2,3,5}
  24:   11000 ~ {4,5}
  28:   11100 ~ {3,4,5}
(End)

Alan Sutcliffe, Divisors and Common Factors in Sets of Integers, awaiting publication

A051026 gives the number of primitive subsets of the integers 1 to n.
The version for prime indices (rather than binary indices) is A316476.
The relatively prime case is A328671.
Partitions with no consecutive divisible parts are A328171.
Compositions without consecutive divisible parts are A328460.
A ranking of antichains is A326704.
Cf. A000120, A048793, A070939, A285572, A285573, A303362, A304713, A305148, A328593.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[0,100],stableQ[Join@@Position[Reverse[IntegerDigits[#,2]],1],Divisible]&] (* Gus Wiseman, Oct 31 2019 *)

cp's OEIS Frontend

A051026 Number of primitive subsequences of {1, 2, ..., n}.

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A323092 Number of double-free integer partitions of n.

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A320340 Heinz numbers of double-free integer partitions.

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A328673 Number of integer partitions of n in which no two distinct parts are relatively prime.

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A319564 Number of T_0 integer partitions of n.

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A200976 Number of partitions of n such that each pair of parts (if any) has a common factor.

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A305149 Number of factorizations of n whose distinct factors are pairwise indivisible and greater than 1.

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A371128 Number of strict integer partitions of n containing all distinct divisors of all parts.

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