A316476 -id:A316476 - cp's OEIS frontend

A319837 Numbers whose distinct prime indices are pairwise indivisible and whose own prime indices span an initial interval of positive integers.

1, 2, 3, 4, 7, 8, 9, 13, 15, 16, 19, 27, 32, 35, 37, 45, 49, 53, 61, 64, 69, 75, 81, 89, 91, 95, 113, 128, 131, 135, 141, 143, 145, 151, 161, 165, 169, 175, 207, 223, 225, 243, 245, 247, 251, 256, 265, 281, 299, 309, 311, 329, 343, 355, 359, 361, 375, 377, 385

Offset: 1

Gus Wiseman, Dec 16 2018

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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of not necessarily strict antichains of multisets spanning an initial interval of positive integers.

			The sequence of multisystems whose MM-numbers belong to the sequence begins:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  13: {{1,2}}
  15: {{1},{2}}
  16: {{},{},{},{}}
  19: {{1,1,1}}
  27: {{1},{1},{1}}
  32: {{},{},{},{},{}}
  35: {{2},{1,1}}
  37: {{1,1,2}}
  45: {{1},{1},{2}}
  49: {{1,1},{1,1}}
  53: {{1,1,1,1}}
  61: {{1,2,2}}
  64: {{},{},{},{},{},{}}
  69: {{1},{2,2}}
  75: {{1},{2},{2}}
  81: {{1},{1},{1},{1}}
  89: {{1,1,1,2}}
  91: {{1,1},{1,2}}
  95: {{2},{1,1,1}}

Cf. A003963, A006126, A055932, A056239, A112798, A285572, A290103, A293993, A302242, A304713, A316476, A318401, A319721, A320275, A320456.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[200],And[normQ[primeMS/@primeMS[#]],stableQ[primeMS[#],Divisible]]&]

A320275 Numbers whose distinct prime indices are pairwise indivisible and whose own prime indices are connected and span an initial interval of positive integers.

Original entry on oeis.org

2, 3, 7, 9, 13, 19, 27, 37, 49, 53, 61, 81, 89, 91, 113, 131, 151, 169, 223, 243, 247, 251, 281, 299, 311, 343, 359, 361, 377, 427, 463, 503, 593, 611, 637, 659, 689, 703, 719, 729, 791, 827, 851, 863, 923, 953, 1069, 1073, 1159, 1163, 1183, 1291, 1321, 1339

Offset: 1

Gus Wiseman, Dec 16 2018

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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of not necessarily strict connected antichains of multisets spanning an initial interval of positive integers.

			The sequence of multisystems whose MM-numbers belong to the sequence begins:
    2: {{}}
    3: {{1}}
    7: {{1,1}}
    9: {{1},{1}}
   13: {{1,2}}
   19: {{1,1,1}}
   27: {{1},{1},{1}}
   37: {{1,1,2}}
   49: {{1,1},{1,1}}
   53: {{1,1,1,1}}
   61: {{1,2,2}}
   81: {{1},{1},{1},{1}}
   89: {{1,1,1,2}}
   91: {{1,1},{1,2}}
  113: {{1,2,3}}
  131: {{1,1,1,1,1}}
  151: {{1,1,2,2}}
  169: {{1,2},{1,2}}

Cf. A003963, A006126, A055932, A056239, A112798, A285572, A286520, A290103, A293994, A302242, A316476, A319496, A319837, A320456, A320532.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[200],And[normQ[primeMS/@primeMS[#]],stableQ[primeMS[#],Divisible],Length[zsm[primeMS[#]]]==1]&]

A322438 Number of unordered pairs of factorizations of n into factors > 1 where no factor of one properly divides any factor of the other.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4

Offset: 1

Gus Wiseman, Dec 08 2018

nonn

First differs from A322437 at a(144) = 4, A322437(144) = 3.

First differs from A379958 at a(120) = 2, A379958(120) = 1.

			The a(240) = 5 pairs of factorizations::
  (4*4*15)|(4*6*10)
    (6*40)|(15*16)
    (8*30)|(12*20)
   (10*24)|(15*16)
   (12*20)|(15*16)

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n.

Cf. A001055, A285572, A285573, A303362, A303386, A304713, A305149, A305150, A305193, A316476, A317144, A322435, A322436, A322437, A322442, A379958.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
divpropQ[x_,y_]:=And[x!=y,Divisible[x,y]];
Table[Length[Select[Subsets[facs[n],{2}],And[!Or@@divpropQ@@@Tuples[#],!Or@@divpropQ@@@Reverse/@Tuples[#]]&]],{n,100}]

factorizations(n, m=n, f=List([]), z=List([])) = if(1==n, listput(z,Vec(f)); z, my(newf); fordiv(n, d, if((d>1)&&(d<=m), newf = List(f); listput(newf,d); z = factorizations(n/d, d, newf, z))); (z));
is_proper_ndf_pair(fac1,fac2) = { for(i=1,#fac1,for(j=1,#fac2,if((fac1[i]!=fac2[j]) && (!(fac1[i]%fac2[j]) || !(fac2[j]%fac1[i])),return(0)))); (1); };
number_of_proper_ndfpairs(z) = sum(i=1,#z,sum(j=i+1,#z,is_proper_ndf_pair(z[i],z[j])));
A322438(n) = number_of_proper_ndfpairs(Vec(factorizations(n))); \\ Antti Karttunen, Jan 24 2025

Data section extended up to a(144) by Antti Karttunen, Jan 24 2025

A338331 Numbers whose set of distinct prime indices (A304038) is pairwise coprime, where a singleton is always considered coprime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Gus Wiseman, Oct 31 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 partitions whose set of distinct parts is a singleton or pairwise coprime. The Heinz number of a 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:
      1: {}          16: {1,1,1,1}     32: {1,1,1,1,1}
      2: {1}         17: {7}           33: {2,5}
      3: {2}         18: {1,2,2}       34: {1,7}
      4: {1,1}       19: {8}           35: {3,4}
      5: {3}         20: {1,1,3}       36: {1,1,2,2}
      6: {1,2}       22: {1,5}         37: {12}
      7: {4}         23: {9}           38: {1,8}
      8: {1,1,1}     24: {1,1,1,2}     40: {1,1,1,3}
      9: {2,2}       25: {3,3}         41: {13}
     10: {1,3}       26: {1,6}         43: {14}
     11: {5}         27: {2,2,2}       44: {1,1,5}
     12: {1,1,2}     28: {1,1,4}       45: {2,2,3}
     13: {6}         29: {10}          46: {1,9}
     14: {1,4}       30: {1,2,3}       47: {15}
     15: {2,3}       31: {11}          48: {1,1,1,1,2}

A302798 is the squarefree case.
A304709 counts partitions with pairwise coprime distinct parts, with ordered version A337665 and Heinz numbers A304711.
A304711 does not consider singletons relatively prime, except for (1).
A304712 counts the partitions with these Heinz numbers.
A316476 is the version for indivisibility instead of relative primality.
A328867 is the pairwise non-coprime instead of pairwise coprime version.
A337600 counts triples of this type, with ordered version A337602.
A338330 is the complement.
A000961 lists powers of primes.
A051424 counts pairwise coprime or singleton partitions.
A304038 gives the distinct prime indices of each positive integer.
A327516 counts pairwise coprime partitions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A000837, A047968, A056239, A112798, A289509, A302797, A305148, A318716, A318719, A337664, A337695.

Select[Range[100],#==1||PrimePowerQ[#]||CoprimeQ@@PrimePi/@First/@FactorInteger[#]&]

Equals A304711 \/ A000961.

A371445 Numbers whose distinct prime indices are binary carry-connected and have no binary containments.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 55, 59, 61, 64, 65, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 128, 131, 137, 139, 143, 145, 149, 151, 157, 163, 167, 169, 173, 179, 181

Offset: 1

Gus Wiseman, Mar 30 2024

nonn

Also Heinz numbers of binary carry-connected integer partitions whose distinct parts have no binary containments, counted by A371446.
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
A binary carry of two positive integers is an overlap of binary indices. A multiset is said to be binary carry-connected iff the graph whose vertices are the elements and whose edges are binary carries is connected.
A binary containment is a containment of binary indices. For example, the numbers {3,5} have binary indices {{1,2},{1,3}}, so there is a binary carry but not a binary containment.

			The terms together with their prime indices begin:
     2: {1}            37: {12}              97: {25}
     3: {2}            41: {13}             101: {26}
     4: {1,1}          43: {14}             103: {27}
     5: {3}            47: {15}             107: {28}
     7: {4}            49: {4,4}            109: {29}
     8: {1,1,1}        53: {16}             113: {30}
     9: {2,2}          55: {3,5}            115: {3,9}
    11: {5}            59: {17}             121: {5,5}
    13: {6}            61: {18}             125: {3,3,3}
    16: {1,1,1,1}      64: {1,1,1,1,1,1}    127: {31}
    17: {7}            65: {3,6}            128: {1,1,1,1,1,1,1}
    19: {8}            67: {19}             131: {32}
    23: {9}            71: {20}             137: {33}
    25: {3,3}          73: {21}             139: {34}
    27: {2,2,2}        79: {22}             143: {5,6}
    29: {10}           81: {2,2,2,2}        145: {3,10}
    31: {11}           83: {23}             149: {35}
    32: {1,1,1,1,1}    89: {24}             151: {36}

Contains all powers of primes A000961 except 1.
Case of A325118 (counted by A325098) without binary containments.
For binary indices of binary indices we have A326750 = A326704 /\ A326749.
For prime indices of prime indices we have A329559 = A305078 /\ A316476.
An opposite version is A371294 = A087086 /\ A371291.
Partitions of this type are counted by A371446.
Carry-connected case of A371455 (counted by A325109).
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A326964 counts connected set-systems, covering A323818.
Cf. A019565, A056239, A112798, A304713, A304716, A305079, A305148, A325097, A325105, A325107, A325119, A371452.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Select[Range[100],stableQ[bpe/@prix[#],SubsetQ] && Length[csm[bpe/@prix[#]]]==1&]

Intersection of A371455 and A325118.

A318730 Number of cyclic compositions (necklaces of positive integers) summing to n with adjacent parts (including the last and first part) being indivisible (either way).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 3, 6, 5, 8, 7, 14, 15, 21, 31, 39, 51, 69, 98, 133, 177, 254, 329, 471, 632, 902, 1230, 1710, 2370, 3270, 4591, 6384, 8898, 12429, 17252, 24230, 33783, 47405, 66254, 92860, 130142, 182469, 256262, 359676, 505231, 710059, 997953, 1404215

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(14) = 14 cyclic compositions with adjacent parts indivisible either way:
  (14)
  (3,11) (4,10) (5,9) (6,8)
  (2,5,7) (2,7,5) (3,4,7) (3,7,4)
  (2,3,2,7) (2,3,4,5) (2,5,2,5) (2,5,4,3) (3,4,3,4)

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A000740, A008965, A059966, A167606, A285573, A303362, A304713, A316476, A318726, A318727, A318728, A318729, A328601.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,And[neckQ[#],And@@Not/@Divisible@@@Partition[#,2,1,1],And@@Not/@Divisible@@@Reverse/@Partition[#,2,1,1]]]&]],{n,20}]

b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q,]}
seq(n)={my(v=sum(k=1, n, k*b(n, k, (i,j)->i%j<>0 && j%i<>0))); vector(n, n, 1 + sumdiv(n, d, eulerphi(d)*v[n/d])/n)} \\ Andrew Howroyd, Oct 27 2019

a(n) = A328601(n) + 1. - Andrew Howroyd, Oct 27 2019

Terms a(21) and beyond from Andrew Howroyd, Sep 08 2018

A328868 Heinz numbers of integer partitions with no two (not necessarily distinct) parts relatively prime, but with no divisor in common to all of the parts.

Original entry on oeis.org

17719, 40807, 43381, 50431, 74269, 83143, 101543, 105703, 116143, 121307, 123469, 139919, 140699, 142883, 171613, 181831, 185803, 191479, 203557, 205813, 211381, 213239, 215267, 219271, 230347, 246703, 249587, 249899, 279371, 286897, 289007, 296993, 300847

Offset: 1

Gus Wiseman, Oct 30 2019

nonn

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

			The sequence of terms together with their prime indices begins:
   17719: {6,10,15}
   40807: {6,14,21}
   43381: {6,15,20}
   50431: {10,12,15}
   74269: {6,10,45}
   83143: {10,15,18}
  101543: {6,21,28}
  105703: {6,15,40}
  116143: {12,14,21}
  121307: {10,15,24}
  123469: {12,15,20}
  139919: {6,15,50}
  140699: {6,22,33}
  142883: {6,10,75}
  171613: {6,14,63}
  181831: {6,20,45}
  185803: {10,14,35}
  191479: {14,18,21}
  203557: {15,18,20}
  205813: {10,15,36}
  211381: {10,12,45}
  213239: {6,15,70}
  215267: {6,10,105}
  219271: {6,26,39}
  230347: {6,6,10,15}

These are the Heinz numbers of the partitions counted by A202425.
Terms of A328679 that are not powers of 2.
The strict case is A318716 (preceded by 2).
A ranking using binary indices (instead of prime indices) is A326912.
Heinz numbers of relatively prime partitions are A289509.
Cf. A000837, A056239, A112798, A200976, A291166, A302796, A316476, A318715, A319752, A319759, A328336, A328672, A328677, A328867.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dv=Select[Range[100000],GCD@@primeMS[#]==1&&And[And@@(GCD[##]>1&)@@@Tuples[Union[primeMS[#]],2]]&]

A329554 Smallest MM-number of a set of n nonempty sets with no singletons.

Original entry on oeis.org

1, 13, 377, 16211, 761917, 55619941, 4393975339, 443791509239, 50148440544007, 6870336354528959, 954976753279525301, 142291536238649269849, 23193520406899830985387, 3873317907952271774559629, 701070541339361191195292849, 139513037726532877047863276951

Offset: 0

Gus Wiseman, Nov 17 2019

nonn

A multiset multisystem is a finite multiset of finite multisets. 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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their corresponding systems begins:
       1: {}
      13: {{1,2}}
     377: {{1,2},{1,3}}
   16211: {{1,2},{1,3},{1,4}}
  761917: {{1,2},{1,3},{1,4},{2,3}}

The smallest BII-number of a set of n sets is A000225(n).
BII-numbers of set-systems with no singletons are A326781.
MM-numbers of sets of nonempty sets are the odd terms of A302494.
MM-numbers of multisets of nonempty non-singleton sets are A320629.
The version with empty edges is A329556.
The version with singletons is A329557.
The version with empty edges and singletons is A329558.
Cf. A056239, A072639, A112798, A279952, A302242, A326031, A329552, A329556.
Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).

sqvs=Select[Range[2,30],SquareFreeQ[#]&&!PrimeQ[#]&];
Table[Times@@Prime/@Take[sqvs,k],{k,0,Length[sqvs]}]

a(n) = Product_{i = 1..n} prime(A120944(i)).

A329556 Smallest MM-number of a set of n sets with no singletons.

Original entry on oeis.org

1, 2, 26, 754, 32422, 1523834

Offset: 0

Gus Wiseman, Nov 17 2019

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 multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their corresponding systems begins:
        1: {}
        2: {{}}
       26: {{},{1,2}}
      754: {{},{1,2},{1,3}}
    32422: {{},{1,2},{1,3},{1,4}}
  1523834: {{},{1,2},{1,3},{1,4},{2,3}}

MM-numbers of sets of sets with no singletons are A329630.
The case without empty edges is A329554.
MM-numbers of sets of sets are A302494.
Cf. A056239, A112798, A302242, A329552, A329557, A329558.
Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dae=Select[Range[100000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&FreeQ[primeMS[#],_?PrimeQ]&];
Table[dae[[Position[PrimeOmega/@dae,k][[1,1]]]],{k,First[Split[Union[PrimeOmega/@dae],#2==#1+1&]]}]

A371446 Number of carry-connected integer partitions whose distinct parts have no binary containments.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 2, 5, 4, 4, 4, 8, 4, 7, 7, 12, 10, 14, 12, 15, 19, 19, 21, 32, 27, 33, 40, 46, 47, 61, 52, 75, 89, 95, 104, 129, 129, 149, 176, 188, 208, 249, 257, 296, 341, 373, 394, 476, 496, 552

Offset: 0

Gus Wiseman, Apr 02 2024

These partitions are ranked by A371445.
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
A binary carry of two positive integers is an overlap of binary indices. An integer partition is binary carry-connected iff the graph with one vertex for each part and edges corresponding to binary carries is connected.
A binary containment is a containment of binary indices. For example, the numbers {3,5} have binary indices {{1,2},{1,3}}, so there is a binary carry but not a binary containment.

			The a(12) = 8 through a(14) = 7 partitions:
  (12)             (13)                         (14)
  (6,6)            (10,3)                       (7,7)
  (9,3)            (5,5,3)                      (9,5)
  (4,4,4)          (1,1,1,1,1,1,1,1,1,1,1,1,1)  (6,5,3)
  (6,3,3)                                       (5,3,3,3)
  (3,3,3,3)                                     (2,2,2,2,2,2,2)
  (2,2,2,2,2,2)                                 (1,1,1,1,1,1,1,1,1,1,1,1,1,1)
  (1,1,1,1,1,1,1,1,1,1,1,1)

The first condition (carry-connected) is A325098.
The second condition (stable) is A325109.
Ranks for binary indices of binary indices are A326750 = A326704 /\ A326749.
Ranks for prime indices of prime indices are A329559 = A305078 /\ A316476.
Ranks for prime indices of binary indices are A371294 = A087086 /\ A371291.
Ranks for binary indices of prime indices are A371445 = A325118 /\ A371455.
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A326964 counts connected set-systems, covering A323818.
Cf. A019565, A304713, A304716, A305079, A305148, A325097, A325105, A325107, A325119, A371452.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]},If[c=={},s, csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n], stableQ[bix/@Union[#],SubsetQ]&&Length[csm[bix/@#]]<=1&]],{n,0,30}]

cp's OEIS Frontend

A319837 Numbers whose distinct prime indices are pairwise indivisible and whose own prime indices span an initial interval of positive integers.

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A320275 Numbers whose distinct prime indices are pairwise indivisible and whose own prime indices are connected and span an initial interval of positive integers.

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A322438 Number of unordered pairs of factorizations of n into factors > 1 where no factor of one properly divides any factor of the other.

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A338331 Numbers whose set of distinct prime indices (A304038) is pairwise coprime, where a singleton is always considered coprime.

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A371445 Numbers whose distinct prime indices are binary carry-connected and have no binary containments.

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A318730 Number of cyclic compositions (necklaces of positive integers) summing to n with adjacent parts (including the last and first part) being indivisible (either way).

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A328868 Heinz numbers of integer partitions with no two (not necessarily distinct) parts relatively prime, but with no divisor in common to all of the parts.

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A329554 Smallest MM-number of a set of n nonempty sets with no singletons.

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