A325102 -id:A325102 - cp's OEIS frontend

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

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

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.

A247935 Number of integer partitions of n whose distinct parts have no binary carries.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 10, 11, 14, 18, 21, 26, 30, 38, 49, 47, 55, 66, 74, 84, 96, 110, 126, 134, 151, 171, 195, 209, 235, 272, 318, 307, 349, 377, 422, 448, 491, 534, 595, 617, 674, 734, 801, 841, 925, 998, 1098, 1118, 1219, 1299, 1418, 1476, 1591, 1711, 1865

Offset: 0

David S. Newman, Sep 26 2014

nonn

From Gus Wiseman, Mar 31 2019: (Start)
A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. For example, the reversed binary expansions of 2, 5, and 8 are
  {0,1}
  {1,0,1}
  {0,0,0,1}
and since there are no columns with more than one 1, the partition (8,5,2) is counted under a(15). The Heinz numbers of these partitions are given by A325097.
(End)

			From _Gus Wiseman_, Mar 30 2019: (Start)
The a(1) = 1 through a(8) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (41)     (33)      (43)       (44)
             (111)  (211)   (221)    (42)      (52)       (422)
                    (1111)  (2111)   (222)     (61)       (611)
                            (11111)  (411)     (421)      (2222)
                                     (2211)    (2221)     (4211)
                                     (21111)   (4111)     (22211)
                                     (111111)  (22111)    (41111)
                                               (211111)   (221111)
                                               (1111111)  (2111111)
                                                          (11111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000110, A000120, A050315, A070939, A080572, A248605, A267610.

Cf. A325095, A325096, A325097, A325098, A325102, A325103, A325109.

with(Bits):
b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, t) +`if`(i>n or And(t, i)>0, 0,
      add(b(n-i*j, i-1, Or(t, i)), j=1..n/i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Dec 28 2014

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[#,Intersection[binpos[#1],binpos[#2]]!={}&]&]],{n,0,20}] (* Gus Wiseman, Mar 30 2019 *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, t] + If[i > n || BitAnd[t, i] > 0, 0, Sum[b[n - i*j, i - 1, BitOr[t, i]], {j, 1, n/i}]]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 80] (* Jean-François Alcover, May 23 2021, after Alois P. Heinz *)

More terms from Alois P. Heinz, Oct 15 2014

Name edited by Gus Wiseman, Mar 31 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.

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.

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]]!={}&]&]

cp's OEIS Frontend

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

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A247935 Number of integer partitions of n whose distinct parts have 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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A325101 Number of divisible binary-containment pairs of positive integers up to n.

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

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