A325123 -id:A325123 - 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)).

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

Original entry on oeis.org

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

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.

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).

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.

A307314 Number of divisors d of 2n such that adding d to 2n in binary requires no carries.

Original entry on oeis.org

1, 2, 1, 3, 2, 3, 1, 4, 2, 3, 1, 5, 1, 2, 1, 5, 2, 5, 1, 5, 2, 2, 1, 7, 2, 2, 2, 4, 1, 3, 1, 6, 2, 4, 1, 7, 2, 3, 1, 7, 2, 4, 1, 3, 2, 2, 1, 9, 1, 4, 2, 3, 1, 4, 1, 6, 1, 2, 1, 6, 1, 2, 1, 7, 4, 6, 1, 6, 2, 3, 1, 10, 2, 3, 1, 4, 1, 3, 1, 9

Offset: 1

Charlie Neder, Apr 02 2019

nonn

Equivalently, number of numbers d such that d|2n and d AND 2n = 0.

First differences of either bisection of A325123.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A325123, A001511, A000005, A247068.

a[n_] := DivisorSum[2*n, 1 &, BitAnd[#, 2*n] == 0 &]; Array[a, 100] (* Amiram Eldar, Feb 08 2025 *)

a(n) = sumdiv(2*n, d, bitand(d, 2*n) == 0); \\ Michel Marcus, Apr 02 2019

A001511(n) <= a(n) <= A000005(n).

a(p) = 1 + [p is in A247068] for p prime, where [] is the Iverson bracket.

cp's OEIS Frontend

A050315 Main diagonal of A050314.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A307314 Number of divisors d of 2n such that adding d to 2n in binary requires no carries.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula