A247935 -id:A247935 - cp's OEIS frontend

A050315 Main diagonal of A050314.

1, 1, 1, 2, 1, 2, 2, 5, 1, 2, 2, 5, 2, 5, 5, 15, 1, 2, 2, 5, 2, 5, 5, 15, 2, 5, 5, 15, 5, 15, 15, 52, 1, 2, 2, 5, 2, 5, 5, 15, 2, 5, 5, 15, 5, 15, 15, 52, 2, 5, 5, 15, 5, 15, 15, 52, 5, 15, 15, 52, 15, 52, 52, 203, 1, 2, 2, 5, 2, 5, 5, 15, 2, 5, 5, 15, 5, 15, 15, 52, 2, 5, 5, 15, 5, 15

Offset: 0

Christian G. Bower, Sep 15 1999

nonn

Also, a(n) is the number of odd multinomial coefficients n!/(k_1!...k_m!) with 1 <= k_1 <= ... <= k_m and k_1 + ... + k_m = n. - Pontus von Brömssen, Mar 23 2018
From Gus Wiseman, Mar 30 2019: (Start)
Also the number of strict integer partitions of n with no binary carries. The Heinz numbers of these partitions are given by A325100. A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. For example, the a(1) = 1 through a(15) = 15 strict integer partitions with no binary carries are:
(1) (2) (3)  (4) (5)  (6)  (7)   (8) (9)  (A)  (B)   (C)  (D)   (E)   (F)
        (21)     (41) (42) (43)      (81) (82) (83)  (84) (85)  (86)  (87)
                           (52)                (92)       (94)  (A4)  (96)
                           (61)                (A1)       (C1)  (C2)  (A5)
                           (421)               (821)      (841) (842) (B4)
                                                                      (C3)
                                                                      (D2)
                                                                      (E1)
                                                                      (843)
                                                                      (852)
                                                                      (861)
                                                                      (942)
                                                                      (A41)
                                                                      (C21)
                                                                      (8421)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..16383
Michael Gilleland, Some Self-Similar Integer Sequences

Cf. A000110, A000120, A050314.
Cf. A070939, A080572, A247935, A267610.
Cf. A325093, A325095, A325096, A325099, A325100, A325103, A325110, A325123.
Main diagonal of A307431 and of A307505.

a:= n-> combinat[bell](add(i,i=convert(n, base, 2))):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 08 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],UnsameQ@@#&&stableQ[#,Intersection[binpos[#1],binpos[#2]]!={}&]&]],{n,0,20}] (* Gus Wiseman, Mar 30 2019 *)
a[n_] := BellB[DigitCount[n, 2, 1]];
a /@ Range[0, 100] (* Jean-François Alcover, May 21 2021 *)

Bell number of number of 1's in binary: a(n) = A000110(A000120(n)).

A080572 Number of ordered pairs (i,j), 0 <= i,j < n, for which (i & j) is nonzero, where & is the bitwise AND operator.

Original entry on oeis.org

0, 0, 1, 2, 7, 8, 15, 24, 37, 38, 49, 62, 81, 98, 121, 146, 175, 176, 195, 216, 247, 272, 307, 344, 387, 420, 463, 508, 559, 608, 663, 720, 781, 782, 817, 854, 909, 950, 1009, 1070, 1141, 1190, 1257, 1326, 1405, 1478, 1561, 1646, 1737, 1802, 1885, 1970, 2065, 2154

Offset: 0

Richard Bean, Feb 22 2003

Conjectured to be less than or equal to lcs(n) (see sequence A063437). The value of a(2^n) is that given in Stinson and van Rees and the value of a(2^n-1) is that given in Fu, Fu and Liao. This function gives an easy way to generate these two constructions.
From Gus Wiseman, Mar 30 2019: (Start)
Also the number of ordered pairs of positive integers up to n with at least one binary carry. A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. For example, the a(2) = 1 through a(6) = 15 ordered pairs are:
  (1,1)  (1,1)  (1,1)  (1,1)  (1,1)
         (2,2)  (1,3)  (1,3)  (1,3)
                (2,2)  (2,2)  (1,5)
                (2,3)  (2,3)  (2,2)
                (3,1)  (3,1)  (2,3)
                (3,2)  (3,2)  (3,1)
                (3,3)  (3,3)  (3,2)
                       (4,4)  (3,3)
                              (3,5)
                              (4,4)
                              (4,5)
                              (5,1)
                              (5,3)
                              (5,4)
                              (5,5)
(End)
a(n) is also the number of even elements in the n X n symmetric Pascal matrix. - Stefano Spezia, Nov 14 2022

C. Fu, H. Fu and W. Liao, A new construction for a critical set in special Latin squares, Proceedings of the Twenty-sixth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, Florida, 1995), Congressus Numerantium, Vol. 110 (1995), pp. 161-166.
D. R. Stinson and G. H. J. van Rees, Some large critical sets, Proceedings of the Eleventh Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, Manitoba, 1981), Congressus Numerantium, Vol. 34 (1982), pp. 441-456.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
R. Bean, Three problems on partial Latin squares, Problem 418 (BCC19,2), Discrete Math., 293 (2005), 314-315.
J. M. Dover, On two OEIS conjectures, arXiv:1606.08033 [math.CO], 2016.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 29.

Cf. A063437.
Cf. A000120, A005061, A006218, A050315, A155609, A247935, A267610, A267700.
Cf. A325096, A325098, A325102, A325103, A325104, A325106, A325124.

f:=proc(n) option remember; local t;
if n <= 1 then 0
elif (n mod 2) =  0 then 3*f(n/2)+(n/2)^2
else t:=(n-1)/2; f(t)+2*f(t+1)+t^2-1; fi; end;
[seq(f(n),n=0..100)]; # N. J. A. Sloane, Jul 01 2017

a[0] = a[1] = 0; a[n_] := a[n] = If[EvenQ[n], 3*a[n/2] + n^2/4, 2*a[(n-1)/2 + 1] + a[(n-1)/2] + (1/4)*(n-1)^2 - 1];
Array[a, 60, 0] (* Jean-François Alcover, Dec 09 2017, from Dover's formula *)
Table[Length[Select[Tuples[Range[n-1],2],Intersection[Position[Reverse[IntegerDigits[#[[1]],2]],1],Position[Reverse[IntegerDigits[#[[2]],2]],1]]!={}&]],{n,0,20}] (* Gus Wiseman, Mar 30 2019 *)

a(2^n) = 4^n-3^n = A005061(n); a(2^n+1) = 4^n-3^n+1 = A155609(n); a(2^n-1) = 4^n-3^n-2^(n+1)+3.
a(0)=a(1)=0, a(2n) = 3a(n)+n^2, a(2n+1) = a(n)+2a(n+1)+n^2-1. This was proved by Jeremy Dover. - Ralf Stephan, Dec 08 2004
a(n) = (A325104(n) - n)/2. - Gus Wiseman, Mar 30 2019

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

A325103 Number of increasing pairs of positive integers up to n with no binary carries.

Original entry on oeis.org

0, 0, 1, 1, 4, 5, 6, 6, 13, 16, 19, 20, 23, 24, 25, 25, 40, 47, 54, 57, 64, 67, 70, 71, 78, 81, 84, 85, 88, 89, 90, 90, 121, 136, 151, 158, 173, 180, 187, 190, 205, 212, 219, 222, 229, 232, 235, 236, 251, 258, 265, 268, 275, 278, 281, 282, 289, 292, 295, 296

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.

			The a(2) = 1 through a(9) = 16 pairs:
  {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}
                {2,4}  {2,4}  {1,6}  {1,6}  {1,6}  {1,6}
                {3,4}  {2,5}  {2,4}  {2,4}  {1,8}  {1,8}
                       {3,4}  {2,5}  {2,5}  {2,4}  {2,4}
                              {3,4}  {3,4}  {2,5}  {2,5}
                                            {2,8}  {2,8}
                                            {3,4}  {2,9}
                                            {3,8}  {3,4}
                                            {4,8}  {3,8}
                                            {5,8}  {4,8}
                                            {6,8}  {4,9}
                                            {7,8}  {5,8}
                                                   {6,8}
                                                   {6,9}
                                                   {7,8}

Cf. A006218, A050315, A080572, A247935, A267610, A267700.

Cf. A325094, A325096, A325098, A325102, A325104, A325106, A325108, A325123.

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

a(n) = A325102(n)/2.

A325104 Number of increasing pairs of positive integers up to n with at least one binary carry.

Original entry on oeis.org

0, 0, 0, 2, 2, 5, 9, 15, 15, 20, 26, 35, 43, 54, 66, 80, 80, 89, 99, 114, 126, 143, 161, 182, 198, 219, 241, 266, 290, 317, 345, 375, 375, 392, 410, 437, 457, 486, 516, 551, 575, 608, 642, 681, 717, 758, 800, 845, 877, 918, 960, 1007, 1051, 1100, 1150, 1203

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.

The version for ordered pairs is A080572.

			The a(3) = 2 through a(8) = 15 pairs:
  {1,3}  {1,3}  {1,3}  {1,3}  {1,3}  {1,3}
  {2,3}  {2,3}  {1,5}  {1,5}  {1,5}  {1,5}
                {2,3}  {2,3}  {1,7}  {1,7}
                {3,5}  {2,6}  {2,3}  {2,3}
                {4,5}  {3,5}  {2,6}  {2,6}
                       {3,6}  {2,7}  {2,7}
                       {4,5}  {3,5}  {3,5}
                       {4,6}  {3,6}  {3,6}
                       {5,6}  {3,7}  {3,7}
                              {4,5}  {4,5}
                              {4,6}  {4,6}
                              {4,7}  {4,7}
                              {5,6}  {5,6}
                              {5,7}  {5,7}
                              {6,7}  {6,7}

Cf. A006218, A050315, A080572, A247935, A267610, A267700.

Cf. A325094, A325096, A325098, A325102, A325103, A325106, A325108, A325123.

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

a(n) = 2 * A080572(n - 2) + n.

A325118 Heinz numbers of binary carry-connected integer partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 15, 16, 17, 19, 20, 22, 23, 25, 27, 29, 30, 31, 32, 34, 37, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 53, 55, 59, 60, 61, 62, 64, 65, 67, 68, 71, 73, 75, 77, 79, 80, 81, 82, 83, 85, 87, 88, 89, 90, 91, 92, 93, 94, 97, 100

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. An integer partition is binary carry-connected if the graph whose vertices are the parts and whose edges are binary carries is connected.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose prime indices are binary carry-connected. 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}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  15: {2,3}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  22: {1,5}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}

Cf. A019565, A048143, A056239, A112798, A247935, A304716, A305078.

Cf. A325098, A325099, A325105, A325109, A325110, A325119.

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]]]]]]]]];
Select[Range[100],Length[csm[binpos/@PrimePi/@First/@FactorInteger[#]]]<=1&]

A325098 Number of binary carry-connected integer partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 7, 13, 15, 23, 27, 42, 50, 72, 88, 125, 153, 211, 258, 349, 430, 569, 698, 914, 1119, 1444, 1765, 2252, 2745, 3470, 4214, 5276, 6387, 7934, 9568, 11800, 14181, 17379, 20818, 25351, 30264, 36668, 43633, 52589, 62394, 74872, 88576, 105818

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. An integer partition is binary carry-connected if the graph whose vertices are the parts and whose edges are binary carries is connected.

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (322)      (44)
                    (31)    (311)    (51)      (331)      (53)
                    (1111)  (11111)  (222)     (511)      (62)
                                     (321)     (3211)     (71)
                                     (3111)    (31111)    (332)
                                     (111111)  (1111111)  (2222)
                                                          (3221)
                                                          (3311)
                                                          (5111)
                                                          (32111)
                                                          (311111)
                                                          (11111111)

Alois P. Heinz, Table of n, a(n) for n = 0..500

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

Cf. A325096, A325099, A325104, A325106, A325108, A325110, A325118, A325119.

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, i, s) option remember; `if`(n=0, `if`(nops(s)>1, 0, 1),
      `if`(i<1, 0, b(n, i-1, s)+ b(n-i, min(i, n-i), g(i, s))))
    end:
a:= n-> b(n$2, {}):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 29 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[IntegerPartitions[n],Length[csm[binpos/@#]]<=1&]],{n,0,20}]
(* Second program: *)
h[n_, s_] := Module[{i, m = n}, Do[m = BitOr[m, i], {i, s}]; {m}];
g[n_, s_] := Function[w, If[w == {}, s ~Union~ {n}, (s ~Complement~ w) ~Union~
    h[n, w]]][Select[s, BitAnd[n, #] > 0&]];
b[n_, i_, s_] := b[n, i, s] = If[n == 0, If[Length[s] > 1, 0, 1],
    If[i < 1, 0, b[n, i - 1, s] + b[n - i, Min[i, n - i], g[i, s]]]];
a[n_] := b[n, n, {}];
a /@ Range[0, 50] (* Jean-François Alcover, May 11 2021, after Alois P. Heinz *)

a(21)-a(48) from Alois P. Heinz, Mar 29 2019

A325095 Number of subsets of {1...n} with no binary carries.

Original entry on oeis.org

1, 2, 4, 5, 10, 12, 14, 15, 30, 35, 40, 42, 47, 49, 51, 52, 104, 119, 134, 139, 154, 159, 164, 166, 181, 186, 191, 193, 198, 200, 202, 203, 406, 458, 510, 525, 577, 592, 607, 612, 664, 679, 694, 699, 714, 719, 724, 726, 778, 793, 808, 813, 828, 833, 838, 840

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. For example, the binary representations of {2,5,8} are:
  2 =   10,
  5 =  101,
  8 = 1000,
and since there are no columns with more than one 1, {2,5,8} is counted under a(8).

			The a(1) = 1 through a(7) = 15 subsets:
  {}   {}     {}     {}       {}       {}       {}
  {1}  {1}    {1}    {1}      {1}      {1}      {1}
       {2}    {2}    {2}      {2}      {2}      {2}
       {1,2}  {3}    {3}      {3}      {3}      {3}
              {1,2}  {4}      {4}      {4}      {4}
                     {1,2}    {5}      {5}      {5}
                     {1,4}    {1,2}    {6}      {6}
                     {2,4}    {1,4}    {1,2}    {7}
                     {3,4}    {2,4}    {1,4}    {1,2}
                     {1,2,4}  {2,5}    {1,6}    {1,4}
                              {3,4}    {2,4}    {1,6}
                              {1,2,4}  {2,5}    {2,4}
                                       {3,4}    {2,5}
                                       {1,2,4}  {3,4}
                                                {1,2,4}

Alois P. Heinz, Table of n, a(n) for n = 0..16383

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

Cf. A325094, A325096, A325097, A325100, A325103, A325104, A325105.

b:= proc(n, t) option remember; `if`(n=0, 1, b(n-1, t)+
     `if`(Bits[And](n, t)=0, b(n-1, Bits[Or](n, t)), 0))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..63);  # 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[#,Intersection[binpos[#1],binpos[#2]]!={}&]&]],{n,0,10}]

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

a(16)-a(55) from Alois P. Heinz, Mar 28 2019

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

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

cp's OEIS Frontend

A050315 Main diagonal of A050314.

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A080572 Number of ordered pairs (i,j), 0 <= i,j < n, for which (i & j) is nonzero, where & is the bitwise AND operator.

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

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A325103 Number of increasing pairs of positive integers up to n with no binary carries.

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A325104 Number of increasing pairs of positive integers up to n with at least one binary carry.

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A325118 Heinz numbers of binary carry-connected integer partitions.

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A325098 Number of binary carry-connected integer partitions of n.

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A325095 Number of subsets of {1...n} with no binary carries.

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