A325107 -id:A325107 - cp's OEIS frontend

A325106 Number of divisible binary-containment pairs of positive integers up to n.

0, 0, 0, 1, 1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 13, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 31, 32, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 48, 49, 50, 51, 52, 53, 56, 57, 58, 61, 63, 64, 65, 66, 67, 70, 71, 72, 77, 77, 78, 79, 80, 81

Offset: 0

Gus Wiseman, Mar 28 2019

nonn

A pair of positive integers is divisible if the first divides the second, and is a binary containment if the positions of 1's in the reversed binary expansion of the first are a subset of the positions of 1's in the reversed binary expansion of the second.

			The a(3) = 1 through a(12) = 8 pairs:
  {1,3}  {1,3}  {1,3}  {1,3}  {1,3}  {1,3}  {1,3}  {1,3}   {1,3}   {1,3}
                {1,5}  {1,5}  {1,5}  {1,5}  {1,5}  {1,5}   {1,5}   {1,5}
                       {2,6}  {1,7}  {1,7}  {1,7}  {1,7}   {1,7}   {1,7}
                              {2,6}  {2,6}  {1,9}  {1,9}   {1,9}   {1,9}
                                            {2,6}  {2,6}   {2,6}   {2,6}
                                                   {2,10}  {1,11}  {1,11}
                                                           {2,10}  {2,10}
                                                                   {4,12}

Cf. A000005, A006218, A080572, A267610, A267700.

Cf. A325094, A325101, A325102, A325107, A325108, A325109, A325110.

Table[Length[Select[Subsets[Range[n],{2}],Divisible[#[[2]],#[[1]]]&&SubsetQ[Position[Reverse[IntegerDigits[#[[2]],2]],1],Position[Reverse[IntegerDigits[#1[[1]],2]],1]]&]],{n,0,30}]

a(n) = A325101(n) - n.

A325096 Number of maximal subsets of {1...n} with no binary carries.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 5, 7, 9, 10, 12, 13, 14, 15, 15, 20, 25, 27, 32, 34, 36, 37, 42, 44, 46, 47, 49, 50, 51, 52, 52, 67, 82, 87, 102, 107, 112, 114, 129, 134, 139, 141, 146, 148, 150, 151, 166, 171, 176, 178, 183, 185, 187, 188, 193, 195, 197, 198, 200, 201

Offset: 0

Gus Wiseman, Mar 27 2019

A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion.

			The a(1) = 1 through a(9) = 7 maximal subsets:
  {1}  {12}  {3}   {34}   {25}   {16}   {7}    {78}    {69}
             {12}  {124}  {34}   {25}   {16}   {168}   {78}
                          {124}  {34}   {25}   {258}   {168}
                                 {124}  {34}   {348}   {249}
                                        {124}  {1248}  {258}
                                                       {348}
                                                       {1248}

Alois P. Heinz, Table of n, a(n) for n = 0..8192

Cf. A000110, A019565, A050315, A080572, A247935, A267610, A267700.

Cf. A325095, A325099, A325100, A325101, A325103, A325104, A325107.

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[n]],stableQ[#,Intersection[binpos[#1],binpos[#2]]!={}&]&]]],{n,0,10}]

a(2^n - 1) = A000110(n).

a(15)-a(61) from Alois P. Heinz, Mar 28 2019

A325110 Number of strict integer partitions of n with no binary containments.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 5, 2, 3, 2, 6, 3, 6, 7, 15, 8, 10, 6, 13, 6, 10, 12, 23, 13, 16, 16, 26, 21, 30, 37, 60, 43, 52, 42, 60, 42, 50, 54, 81, 59, 60, 66, 80, 74, 86, 108, 145, 119, 125, 126, 144, 134, 140, 170, 208, 189, 193, 221, 248, 253, 292, 323, 435, 392

Offset: 0

Gus Wiseman, Mar 28 2019

nonn

A pair of positive integers is a binary containment if the positions of 1's in the reversed binary expansion of the first are a subset of the positions of 1's in the reversed binary expansion of the second.

			The a(1) = 1 through a(12) = 3 partitions (A = 10, B = 11, C = 12):
  (1)  (2)  (3)   (4)  (5)   (6)   (7)    (8)   (9)   (A)   (B)    (C)
            (21)       (41)  (42)  (43)   (53)  (63)  (82)  (65)   (84)
                                   (52)         (81)        (83)   (93)
                                   (61)                     (92)
                                   (421)                    (A1)
                                                            (821)

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

Cf. A019565, A050315, A267610, A267700.

Cf. A325101, A325106, A325107, A325108, A325109, A325119.

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&stableQ[#,SubsetQ[binpos[#1],binpos[#2]]&]&]],{n,0,30}]

A325109 Number of integer partitions of n whose distinct parts have no binary containments.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 10, 12, 15, 18, 23, 28, 32, 41, 52, 57, 66, 76, 90, 99, 117, 131, 157, 172, 194, 216, 255, 276, 313, 358, 410, 447, 511, 546, 630, 677, 750, 818, 945, 990, 1108, 1200, 1338, 1429, 1606, 1713, 1928, 2062, 2263, 2412, 2725, 2847, 3142, 3389

Offset: 0

Gus Wiseman, Mar 28 2019

nonn

A pair of positive integers is a binary containment if the positions of 1's in the reversed binary expansion of the first are a subset of the positions of 1's in the reversed binary expansion of the second.

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (41)     (33)      (43)       (44)
             (111)  (211)   (221)    (42)      (52)       (53)
                    (1111)  (2111)   (222)     (61)       (422)
                            (11111)  (411)     (421)      (611)
                                     (2211)    (2221)     (2222)
                                     (21111)   (4111)     (4211)
                                     (111111)  (22111)    (22211)
                                               (211111)   (41111)
                                               (1111111)  (221111)
                                                          (2111111)
                                                          (11111111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..400, (terms up to a(200) from Alois P. Heinz)

Cf. A019565, A247935, A267610, A267700.

Cf. A325098, A325101, A325106, A325107, A325108, A325110.

c:= proc() option remember; local i, x, y;
      x, y:= map(n-> Bits[Split](n), [args])[];
      for i to nops(x) do
        if x[i]=1 and y[i]=0 then return false fi
      od; true
    end:
b:= proc(n, i, s) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, s)+`if`(ormap(j-> c(i, j), s), 0, add(
      b(n-i*j, i-1, s union {i}), j=1..n/i))))
    end:
a:= n-> b(n$2, {}):
seq(a(n), n=0..55);  # Alois P. Heinz, Mar 29 2019

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],stableQ[#,SubsetQ[binpos[#1],binpos[#2]]&]&]],{n,0,15}]
(* Second program: *)
c[args_List] := c[args] = Module[{i, x, y}, {x, y} = Reverse@IntegerDigits[#, 2]& /@ args; For[i = 1, i <= Length[x], i++, If[x[[i]] == 1 && y[[i]] == 0, Return[False]]]; True];
b[n_, i_, s_List] := b[n, i, s] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, s] + If[AnyTrue[s, c[{i, #}]&], 0, Sum[b[n - i*j, i-1, s ~Union~ {i}], {j, 1, n/i}]]]];
a[n_] := b[n, n, {}];
a /@ Range[0, 55] (* Jean-François Alcover, Jun 03 2021, after Alois P. Heinz *)

a(31)-a(54) from Alois P. Heinz, Mar 29 2019

A325101 Number of divisible binary-containment pairs of positive integers up to n.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 9, 11, 12, 14, 16, 18, 20, 22, 24, 28, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 53, 55, 57, 61, 63, 64, 66, 68, 70, 72, 74, 76, 79, 81, 83, 85, 87, 89, 93, 95, 97, 99, 101, 103, 107, 109, 111, 115, 118, 120, 122, 124, 126, 130, 132, 134

Offset: 0

Gus Wiseman, Mar 28 2019

nonn

A pair of positive integers is divisible if the first divides the second, and is a binary containment if the positions of 1's in the reversed binary expansion of the first are a subset of those in the second.

			The a(1) = 1 through a(8) = 12 pairs:
  (1,1)  (1,1)  (1,1)  (1,1)  (1,1)  (1,1)  (1,1)  (1,1)
         (2,2)  (1,3)  (1,3)  (1,3)  (1,3)  (1,3)  (1,3)
                (2,2)  (2,2)  (1,5)  (1,5)  (1,5)  (1,5)
                (3,3)  (3,3)  (2,2)  (2,2)  (1,7)  (1,7)
                       (4,4)  (3,3)  (2,6)  (2,2)  (2,2)
                              (4,4)  (3,3)  (2,6)  (2,6)
                              (5,5)  (4,4)  (3,3)  (3,3)
                                     (5,5)  (4,4)  (4,4)
                                     (6,6)  (5,5)  (5,5)
                                            (6,6)  (6,6)
                                            (7,7)  (7,7)
                                                   (8,8)

Cf. A000005, A006218, A080572, A267610, A267700.

Cf. A325094, A325102, A325106, A325107, A325108, A325109, A325110.

Table[Length[Select[Tuples[Range[n],2],Divisible[#[[2]],#[[1]]]&&SubsetQ[Position[Reverse[IntegerDigits[#[[2]],2]],1],Position[Reverse[IntegerDigits[#1[[1]],2]],1]]&]],{n,0,30}]

a(n) = A325106(n) + n.

A325105 Number of binary carry-connected subsets of {1...n}.

Original entry on oeis.org

1, 2, 3, 7, 8, 20, 48, 112, 113, 325, 777, 1737, 3709, 7741, 15869, 32253, 32254, 96538, 225798, 485702, 1006338, 2049602, 4137346, 8315266, 16697102, 33465934, 67007886, 134100366, 268301518, 536720590, 1073575118, 2147316942, 2147316943, 6441886323

Offset: 0

Gus Wiseman, Mar 28 2019

nonn

A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. A subset is binary carry-connected if the graph whose vertices are the elements and whose edges are binary carries is connected.

			The a(0) = 1 through a(4) = 8 subsets:
  {}  {}   {}   {}       {}
      {1}  {1}  {1}      {1}
           {2}  {2}      {2}
                {3}      {3}
                {1,3}    {4}
                {2,3}    {1,3}
                {1,2,3}  {2,3}
                         {1,2,3}

Alois P. Heinz, Table of n, a(n) for n = 0..1023

Cf. A019565, A080572, A247935, A304714, A304716, A305078.
Cf. A325095, A325098, A325099, A325104, A325107, A325118, A325119.
Partial sums of A306299.

h:= proc(n, s) local i, m; m:= n;
      for i in s do m:= Bits[Or](m, i) od; {m}
    end:
g:= (n, s)-> (w-> `if`(w={}, s union {n}, s minus w union
              h(n, w)))(select(x-> Bits[And](n, x)>0, s)):
b:= proc(n, s) option remember; `if`(n=0,
      `if`(nops(s)>1, 0, 1), b(n-1, s)+b(n-1, g(n, s)))
    end:
a:= n-> b(n, {}):
seq(a(n), n=0..35);  # Alois P. Heinz, Mar 31 2019

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Range[n]],Length[csm[binpos/@#]]<=1&]],{n,0,10}]

a(n) = A306297(n,0) + A306297(n,1). - Alois P. Heinz, Mar 31 2019

a(16)-a(33) from Alois P. Heinz, Mar 31 2019

A325108 Number of maximal subsets of {1...n} with no binary containments.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 6, 6, 11, 13, 16, 17, 22, 27, 28

Offset: 0

Gus Wiseman, Mar 28 2019

A pair of positive integers is a binary containment if the positions of 1's in the reversed binary expansion of the first are a subset of the positions of 1's in the reversed binary expansion of the second.

			The a(0) = 1 through a(7) = 6 maximal subsets:
  {}  {1}  {1,2}  {3}    {3,4}    {2,5}    {1,6}    {7}
                  {1,2}  {1,2,4}  {3,4}    {2,5}    {1,6}
                                  {3,5}    {3,4}    {2,5}
                                  {1,2,4}  {1,2,4}  {3,4}
                                           {3,5,6}  {1,2,4}
                                                    {3,5,6}

Cf. A006126, A014466, A019565, A267610.

Cf. A325095, A325096, A325101, A325106, A325107, A325109, A325110.

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[n]],stableQ[#,SubsetQ[binpos[#1],binpos[#2]]&]&]]],{n,0,10}]

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.

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}]

A371455 Numbers k such that if we take the binary indices of each prime index of k we get an antichain of sets.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 35, 36, 37, 38, 41, 42, 43, 47, 48, 49, 52, 53, 54, 55, 56, 57, 58, 59, 61, 63, 64, 65, 67, 69, 71, 72, 73, 74, 76, 79, 81, 83, 84, 86, 89, 95, 96, 97, 98, 99

Offset: 1

Gus Wiseman, Apr 01 2024

nonn

In an antichain of sets, no edge is a proper subset of any other.

			The prime indices of 65 are {3,6} with binary indices {{1,2},{2,3}} so 65 is in the sequence.
The prime indices of 255 are {2,3,7} with binary indices {{2},{1,2},{1,2,3}} so 255 is not in the sequence.

Jakub Buczak, Table of n, a(n) for n = 1..10000

Contains all powers of primes A000961.
An opposite version is A087086, carry-connected case A371294.
For prime indices of prime indices we have A316476, carry-connected A329559.
These antichains are counted by A325109.
For binary indices of binary indices we have A326704, carry-conn. A326750.
The carry-connected case is A371445, counted by A371446.
A048143 counts connected antichains of sets.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A050320 counts set multipartitions of prime indices, see also A318360.
A070939 gives length of binary expansion.
A089259 counts set multipartitions of integer partitions.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A116540 counts normal set multipartitions.
A302478 ranks set multipartitions, cf. A073576.
A325118 ranks carry-connected partitions, counted by A325098.
A371451 counts carry-connected components of binary indices.
Cf. A019565, A050326, A304713, A305148, A325097, A325107, A371291, 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}]]]];
Select[Range[100],stableQ[bix/@prix[#],SubsetQ]&]

cp's OEIS Frontend

A325106 Number of divisible binary-containment pairs of positive integers up to n.

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A325096 Number of maximal subsets of {1...n} with no binary carries.

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A325110 Number of strict integer partitions of n with no binary containments.

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A325109 Number of integer partitions of n whose distinct parts have no binary containments.

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A325101 Number of divisible binary-containment pairs of positive integers up to n.

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A325105 Number of binary carry-connected subsets of {1...n}.

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A325108 Number of maximal subsets of {1...n} with no binary containments.

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

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