A325101 -id:A325101 - cp's OEIS frontend

A267610 Total number of OFF (white) cells after n iterations of the "Rule 182" elementary cellular automaton starting with a single ON (black) cell.

0, 0, 2, 2, 4, 6, 12, 12, 14, 16, 22, 24, 30, 36, 50, 50, 52, 54, 60, 62, 68, 74, 88, 90, 96, 102, 116, 122, 136, 150, 180, 180, 182, 184, 190, 192, 198, 204, 218, 220, 226, 232, 246, 252, 266, 280, 310, 312, 318, 324, 338, 344, 358, 372, 402, 408, 422, 436

Offset: 0

Robert Price, Jan 18 2016

From Gus Wiseman, Mar 30 2019: (Start)
It appears that a(n) is also the number of increasing binary-containment pairs of distinct positive integers up to n + 1. 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. For example, the a(2) = 2 through a(8) = 14 pairs are:
  {1,3}  {1,3}  {1,3}  {1,3}  {1,3}  {1,3}  {1,3}
  {2,3}  {2,3}  {1,5}  {1,5}  {1,5}  {1,5}  {1,5}
                {2,3}  {2,3}  {1,7}  {1,7}  {1,7}
                {4,5}  {2,6}  {2,3}  {2,3}  {1,9}
                       {4,5}  {2,6}  {2,6}  {2,3}
                       {4,6}  {2,7}  {2,7}  {2,6}
                              {3,7}  {3,7}  {2,7}
                              {4,5}  {4,5}  {3,7}
                              {4,6}  {4,6}  {4,5}
                              {4,7}  {4,7}  {4,6}
                              {5,7}  {5,7}  {4,7}
                              {6,7}  {6,7}  {5,7}
                                            {6,7}
                                            {8,9}
(End)

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A000120, A006046, A006218, A070939, A071038, A080572, A267700.

Cf. A325101, A325103, A325104, A325106, A325109, A325110, A325123, A325124.

rule=182; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) nbc=Table[Total[catri[[k]]],{k,1,rows}]; (* Number of Black cells in stage n *) nwc=Table[Length[catri[[k]]]-nbc[[k]],{k,1,rows}]; (* Number of White cells in stage n *) Table[Total[Take[nwc,k]],{k,1,rows}] (* Number of White cells through stage n *)

Conjecture: a(n) = A267700(n) - n. - Gus Wiseman, Mar 30 2019

G.f.: (1/x)*(A(x)/x - (x+1)/(1-x)^2) where A(x) is the g.f. for A006046 (conjectured). - John Tyler Rascoe, Jul 08 2024

A267700 "Tree" sequence in a 90-degree sector of the cellular automaton of A160720.

Original entry on oeis.org

0, 1, 2, 5, 6, 9, 12, 19, 20, 23, 26, 33, 36, 43, 50, 65, 66, 69, 72, 79, 82, 89, 96, 111, 114, 121, 128, 143, 150, 165, 180, 211, 212, 215, 218, 225, 228, 235, 242, 257, 260, 267, 274, 289, 296, 311, 326, 357, 360, 367, 374, 389, 396, 411, 426, 457, 464, 479, 494, 525, 540, 571, 602, 665, 666, 669, 672, 679, 682, 689

Offset: 0

Omar E. Pol, Jan 19 2016

nonn

Conjecture: this is also the "tree" sequence in a 120-degree sector of the cellular automaton of A266532.
It appears that this is also the partial sums of A038573.
a(n) is also the total number of ON cells after n-th stage in the tree that arises from one of the four spokes in a 90-degree sector of the cellular automaton A160720 on the square grid.
Note that the structure of A160720 is also the "outward" version of the Ulam-Warburton cellular automaton of A147562.
It appears that A038573 gives the number of cells turned ON at n-th stage.
Conjecture: a(n) is also the total number of Y-toothpicks after n-th stage in the tree that arises from one of the three spokes in a 120-degree sector of the cellular automaton of A266532 on the triangular grid.
Note that the structure of A266532 is also the "outward" version of the Y-toothpick cellular automaton of A160120.
It appears that A038573 also gives the number of Y-toothpicks added at n-th stage.
Comment from N. J. A. Sloane, Jan 23 2016: All the above conjectures are true!
From Gus Wiseman, Mar 31 2019: (Start)
a(n) is also the number of nondecreasing binary-containment pairs of positive integers up to n. 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. For example, the a(1) = 1 through a(6) = 12 pairs are:
  (1,1)  (1,1)  (1,1)  (1,1)  (1,1)  (1,1)
         (2,2)  (1,3)  (1,3)  (1,3)  (1,3)
                (2,2)  (2,2)  (1,5)  (1,5)
                (2,3)  (2,3)  (2,2)  (2,2)
                (3,3)  (3,3)  (2,3)  (2,3)
                       (4,4)  (3,3)  (2,6)
                              (4,4)  (3,3)
                              (4,5)  (4,4)
                              (5,5)  (4,5)
                                     (4,6)
                                     (5,5)
                                     (6,6)
(End)

David Applegate, The movie version
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata
Index entries for sequences related to toothpick sequences

Cf. A000120, A038573, A147562, A160120, A160720, A161336, A169779, A266532, A266534, A266536.
Cf. A006218, A019565, A070939, A080572, A267610, A267700.
Cf. A325101, A325103, A325104, A325106, A325109, A325110, A325124.

Accumulate[Table[2^DigitCount[n,2,1]-1,{n,0,30}]] (* based on conjecture confirmed by Sloane, Gus Wiseman, Mar 31 2019 *)

a(n) = (A160720(n+1) - 1)/4.
Conjecture 1: a(n) = (A266532(n+1) - 1)/3.
Conjecture 2: a(n) = A160720(n+1) - A266532(n+1).
All of the above conjectures are true. -  N. J. A. Sloane, Jan 23 2016
(Conjecture) a(n) = A267610(n) + n. - Gus Wiseman, Mar 31 2019

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

Original entry on oeis.org

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

A325107 Number of subsets of {1...n} with no binary containments.

Original entry on oeis.org

1, 2, 4, 5, 10, 13, 18, 19, 38, 52, 77, 83, 133, 147, 166, 167, 334, 482, 764, 848, 1465, 1680, 1987, 2007, 3699, 4413, 5488, 5572, 7264, 7412, 7579, 7580, 15160, 22573, 37251, 42824, 77387, 92863, 116453, 118461, 227502, 286775, 382573, 392246, 555661, 574113

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(0) = 1 through a(6) = 18 subsets:
  {}  {}   {}     {}     {}       {}       {}
      {1}  {1}    {1}    {1}      {1}      {1}
           {2}    {2}    {2}      {2}      {2}
           {1,2}  {3}    {3}      {3}      {3}
                  {1,2}  {4}      {4}      {4}
                         {1,2}    {5}      {5}
                         {1,4}    {1,2}    {6}
                         {2,4}    {1,4}    {1,2}
                         {3,4}    {2,4}    {1,4}
                         {1,2,4}  {2,5}    {1,6}
                                  {3,4}    {2,4}
                                  {3,5}    {2,5}
                                  {1,2,4}  {3,4}
                                           {3,5}
                                           {3,6}
                                           {5,6}
                                           {1,2,4}
                                           {3,5,6}

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..129, (terms up to a(71) from Alois P. Heinz)

Cf. A006126, A014466, A019565, A267610.

Cf. A325095, A325096, A325101, A325105, A325106, A325108, A325109, 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, s) option remember; `if`(n=0, 1, b(n-1, s)+
     `if`(ormap(i-> c(n, i), s), 0, b(n-1, s union {n})))
    end:
a:= n-> b(n, {}):
seq(a(n), n=0..34);  # Alois P. Heinz, Mar 28 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[Subsets[Range[n]],stableQ[#,SubsetQ[binpos[#1],binpos[#2]]&]&]],{n,0,13}]

a(2^n - 1) = A014466(n).

a(16)-a(45) from Alois P. Heinz, Mar 28 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}]

A325123 Number of divisible pairs of positive integers up to n with no binary carries.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 4, 4, 7, 7, 9, 9, 12, 12, 13, 13, 17, 17, 19, 19, 22, 22, 23, 23, 28, 28, 29, 29, 31, 31, 32, 32, 37, 37, 39, 39, 44, 44, 45, 45, 50, 50, 52, 52, 54, 54, 55, 55, 62, 62, 64, 64, 66, 66, 68, 68, 72, 72, 73, 73, 76, 76, 77, 77, 83, 83, 85, 85

Offset: 0

Gus Wiseman, Mar 29 2019

nonn

Two positive integers are divisible if the first divides the second, and they have a binary carry if the positions of 1's in their reversed binary expansion overlap.

a(2k+1) = a(2k), since an odd number and any divisor will overlap in the last digit. Additionally, a(2k+2) > a(2k+1) because the pair {1,2k+2} is always valid. Therefore, every term appears exactly twice. - Charlie Neder, Apr 02 2019

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

Cf. A006218, A019565, A050315, A070939, A080572, A247935, A267610.

Cf. A325095, A325096, A325101, A325103, A325104, A325105, A325106, A325124.

Table[Length[Select[Tuples[Range[n],2],Divisible@@Reverse[#]&&Intersection[Position[Reverse[IntegerDigits[#[[1]],2]],1],Position[Reverse[IntegerDigits[#[[2]],2]],1]]=={}&]],{n,0,20}]

A325100 Heinz numbers of strict integer partitions with no binary carries.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 31, 33, 35, 37, 38, 41, 42, 43, 47, 53, 57, 58, 59, 61, 67, 69, 71, 73, 74, 79, 83, 86, 89, 95, 97, 101, 103, 106, 107, 109, 111, 113, 114, 122, 123, 127, 131, 133, 137, 139, 142, 149, 151, 157, 158, 159

Offset: 1

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.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are squarefree numbers whose prime indices have no carries. 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 sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   6: {1,2}
   7: {4}
  11: {5}
  13: {6}
  14: {1,4}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  26: {1,6}
  29: {10}
  31: {11}
  33: {2,5}
  35: {3,4}
  37: {12}
  38: {1,8}
  41: {13}
  42: {1,2,4}

Cf. A050315, A056239, A080572, A112798, A247935, A267610.

Cf. A325095, A325096, A325097, A325100, A325101, A325103, A325110, 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}];
Select[Range[100],SquareFreeQ[#]&&stableQ[PrimePi/@First/@FactorInteger[#],Intersection[binpos[#1],binpos[#2]]!={}&]&]

cp's OEIS Frontend

A267610 Total number of OFF (white) cells after n iterations of the "Rule 182" elementary cellular automaton starting with a single ON (black) cell.

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A267700 "Tree" sequence in a 90-degree sector of the cellular automaton of A160720.

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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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A325107 Number of subsets of {1...n} with no binary containments.

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