A080572 -id:A080572 - cp's OEIS frontend

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

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

A325097 Heinz numbers of integer partitions whose distinct parts have no binary carries.

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, 56, 57, 58, 59, 61, 63, 64, 67, 69, 71, 72, 73, 74, 76, 79, 81, 83, 84, 86, 89, 95, 96, 97, 98, 99, 101

Offset: 1

Gus Wiseman, Mar 27 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 numbers whose distinct prime indices have no binary carries.

			Most small numbers are in the sequence, however the sequence of non-terms together with their prime indices begins:
  10: {1,3}
  15: {2,3}
  20: {1,1,3}
  22: {1,5}
  30: {1,2,3}
  34: {1,7}
  39: {2,6}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}
  46: {1,9}
  50: {1,3,3}
  51: {2,7}
  55: {3,5}
  60: {1,1,2,3}
  62: {1,11}
  65: {3,6}
  66: {1,2,5}
  68: {1,1,7}
  70: {1,3,4}

Cf. A000110, A000720, A001222, A050315, A056239, A080572, A112798, A247935.

Cf. A325094, A325095, A325096, A325097, A325098, A325100, A325102, A325103.

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],stableQ[PrimePi/@First/@FactorInteger[#],Intersection[binpos[#1],binpos[#2]]!={}&]&]

A325102 Number of ordered pairs of positive integers up to n with no binary carries.

Original entry on oeis.org

0, 0, 2, 2, 8, 10, 12, 12, 26, 32, 38, 40, 46, 48, 50, 50, 80, 94, 108, 114, 128, 134, 140, 142, 156, 162, 168, 170, 176, 178, 180, 180, 242, 272, 302, 316, 346, 360, 374, 380, 410, 424, 438, 444, 458, 464, 470, 472, 502, 516, 530, 536, 550, 556, 562, 564, 578

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

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

Cf. A325094, A325096, A325098, A325103, A325104, A325106, A325108, A325123.

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

a(n) = 2 * A325103(n).

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}]

A325099 Number of binary carry-connected strict integer partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 1, 4, 5, 8, 6, 11, 11, 15, 13, 18, 20, 30, 29, 43, 49, 68, 66, 84, 94, 125, 131, 165, 184, 237, 251, 291, 315, 383, 408, 486, 536, 663, 714, 832, 912, 1104, 1195, 1405, 1554, 1877, 2046, 2348, 2559, 2998, 3256, 3730, 4084, 4793, 5230, 5938

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(11) = 6 strict partitions (A = 10, B = 11):
  (1)  (2)  (3)  (4)   (5)   (6)    (7)  (8)   (9)    (A)    (B)
                 (31)  (32)  (51)        (53)  (54)   (64)   (65)
                             (321)       (62)  (63)   (73)   (74)
                                         (71)  (72)   (91)   (632)
                                               (531)  (532)  (731)
                                                      (541)  (5321)
                                                      (631)
                                                      (721)

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

Cf. A325095, A325096, A325098, A325104, A325106, 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]]]]]]]]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[csm[binpos/@#]]<=1&]],{n,0,30}]

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]]!={}&]&]

A325124 Number of divisible pairs of positive integers up to n with at least one binary carry.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 10, 12, 13, 16, 18, 20, 23, 25, 28, 32, 33, 35, 39, 41, 44, 48, 51, 53, 56, 59, 62, 66, 70, 72, 79, 81, 82, 86, 88, 92, 96, 98, 101, 105, 108, 110, 116, 118, 122, 128, 131, 133, 136, 139, 143, 147, 151, 153, 159, 163, 167, 171, 174, 176, 185

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.

			The a(1) = 1 through a(8) = 13 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)  (3,6)  (3,3)  (3,3)
                                     (4,4)  (3,6)  (3,6)
                                     (5,5)  (4,4)  (4,4)
                                     (6,6)  (5,5)  (5,5)
                                            (6,6)  (6,6)
                                            (7,7)  (7,7)
                                                   (8,8)

Cf. A006218, A019565, A050315, A070939, A080572, A247935, A267610, A307230.

Cf. A325095, A325096, A325101, A325103, A325104, A325105, A325106, A325123.

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}]

a(n) = A307230(n) + n.

A307230 Number of divisible pairs of distinct positive integers up to n with at least one binary carry.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 5, 7, 8, 9, 11, 12, 14, 17, 17, 18, 21, 22, 24, 27, 29, 30, 32, 34, 36, 39, 42, 43, 49, 50, 50, 53, 54, 57, 60, 61, 63, 66, 68, 69, 74, 75, 78, 83, 85, 86, 88, 90, 93, 96, 99, 100, 105, 108, 111, 114, 116, 117, 125, 126, 128, 133, 133

Offset: 0

Gus Wiseman, Mar 29 2019

nonn

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

			The a(3) = 1 through a(12) = 11 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}
                       {3,6}  {2,6}  {2,6}  {1,9}  {1,9}   {1,9}   {1,9}
                              {3,6}  {3,6}  {2,6}  {2,6}   {2,6}   {2,6}
                                            {3,6}  {3,6}   {3,6}   {3,6}
                                            {3,9}  {3,9}   {3,9}   {3,9}
                                                   {2,10}  {1,11}  {1,11}
                                                           {2,10}  {2,10}
                                                                   {4,12}
                                                                   {6,12}

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

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

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

a(n) = A325124(n) - n.

cp's OEIS Frontend

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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A325097 Heinz numbers of integer partitions whose distinct parts have no binary carries.

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A325102 Number of ordered pairs of positive integers up to n with no binary carries.

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

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A325099 Number of binary carry-connected strict integer partitions of n.

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

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A325124 Number of divisible pairs of positive integers up to n with at least one binary carry.

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