A265705 -id:A265705 - cp's OEIS frontend

A265736 Row sums of triangle A265705.

0, 2, 8, 12, 29, 38, 46, 56, 107, 126, 144, 164, 177, 198, 218, 240, 407, 446, 484, 524, 557, 598, 638, 680, 691, 734, 776, 820, 857, 902, 946, 992, 1583, 1662, 1740, 1820, 1893, 1974, 2054, 2136, 2187, 2270, 2352, 2436, 2513, 2598, 2682, 2768, 2727, 2814

Offset: 0

Reinhard Zumkeller, Dec 15 2015

nonn

a(A000225(n)) = A020522(n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A265705, A000225, A020522.

Haskell
```
a265736 = sum . a265705_row
```

A053644 Most significant bit of n, msb(n); largest power of 2 less than or equal to n; write n in binary and change all but the first digit to zero.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64

Offset: 0

Henry Bottomley, Mar 22 2000

Except for the initial term, 2^n appears 2^n times. - Lekraj Beedassy, May 26 2005
a(n) is the smallest k such that row k in triangle A265705 contains n. - Reinhard Zumkeller, Dec 17 2015
a(n) is the sum of totient function over powers of 2 <= n. - Anthony Browne, Jun 17 2016
Given positive n, reverse the bits of n and divide by 2^floor(log_2 n). Numerators are in A030101. Ignoring the initial 0, denominators are in this sequence. - Alonso del Arte, Feb 11 2020

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

See A000035 for least significant bit(n).
MASKTRANS transform of A055975 (prepended with 0), MASKTRANSi transform of A048678.
Bisection of A065267, A065279, A065291, A072376.
First differences of A063915. Cf. A076877, A073121.
This is Guy Steele's sequence GS(5, 5) (see A135416).
Equals for n >= 1 the first right hand column of A160464. - Johannes W. Meijer, May 24 2009
Diagonal of A088370. - Alois P. Heinz, Oct 28 2011
Cf. A265705, A000010.

a053644 n = if n <= 1 then n else 2 * a053644 (div n 2)
-- Reinhard Zumkeller, Aug 28 2014
a053644_list = 0 : concat (iterate (\zs -> map (* 2) (zs ++ zs)) [1])
-- Reinhard Zumkeller, Dec 08 2012, Oct 21 2011, Oct 17 2010

[0] cat [2^Ilog2(n): n in [1..90]]; // Vincenzo Librandi, Dec 11 2018

a:= n-> 2^ilog2(n):
seq(a(n), n=0..80);  # Alois P. Heinz, Dec 20 2016

A053644[n_] := 2^(Length[ IntegerDigits[n, 2]] - 1); A053644[0] = 0; Table[A053644[n], {n, 0, 74}] (* Jean-François Alcover, Dec 01 2011 *)
nv[n_] := Module[{c = 2^n}, Table[c, {c}]]; Join[{0}, Flatten[Array[nv, 7, 0]]] (* Harvey P. Dale, Jul 17 2012 *)

a(n)=my(k=1);while(k<=n,k<<=1);k>>1 \\ Charles R Greathouse IV, May 27 2011

a(n) = if(!n, 0, 2^exponent(n)) \\ Iain Fox, Dec 10 2018

def a(n): return 0 if n==0 else 2**(len(bin(n)[2:]) - 1) # Indranil Ghosh, May 25 2017

def A053644(n): return 1<Chai Wah Wu, Jul 27 2022

(0 to 127).map(Integer.highestOneBit()) // _Alonso del Arte, Feb 26 2020

a(n) = a(floor(n / 2)) * 2.
a(n) = 2^A000523(n).
From n >= 1 onward, A053644(n) = A062383(n)/2.
a(0) = 0, a(1) = 1 and a(n+1) = a(n)*floor(n/a(n)). - Benoit Cloitre, Aug 17 2002
G.f.: 1/(1 - x) * (x + Sum_{k >= 1} 2^(k - 1)*x^2^k). - Ralf Stephan, Apr 18 2003
a(n) = (A003817(n) + 1)/2 = A091940(n) + 1. - Reinhard Zumkeller, Feb 15 2004
a(n) = Sum_{k = 1..n} (floor(2^k/k) - floor((2^k - 1)/k))*A000010(k). - Anthony Browne, Jun 17 2016
a(2^m+k) = 2^m, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Aug 07 2016

A089633 Numbers having no more than one 0 in their binary representation.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 11, 13, 14, 15, 23, 27, 29, 30, 31, 47, 55, 59, 61, 62, 63, 95, 111, 119, 123, 125, 126, 127, 191, 223, 239, 247, 251, 253, 254, 255, 383, 447, 479, 495, 503, 507, 509, 510, 511, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1022, 1023

Offset: 0

Reinhard Zumkeller, Jan 01 2004

Complement of A158582. - Reinhard Zumkeller, Apr 16 2009
Also union of A168604 and A030130. - Douglas Latimer, Jul 19 2012
Numbers of the form 2^t - 2^k - 1, 0 <= k < t.
n is in the sequence if and only if 2*n+1 is in the sequence. - Robert Israel, Dec 14 2018
Also the least binary rank of a strict integer partition of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1). - Gus Wiseman, May 24 2024

			From _Tilman Piesk_, May 09 2012: (Start)
This may also be viewed as a triangle:             In binary:
                  0                                         0
               1     2                                 01       10
             3    5    6                          011      101      110
           7   11   13   14                  0111     1011     1101     1110
        15   23   27   29   30          01111    10111    11011    11101    11110
      31  47   55   59   61   62
   63   95  111  119  123  125  126
Left three diagonals are A000225,  A055010, A086224. Right diagonal is A000918. Central column is A129868. Numbers in row n (counted from 0) have n binary 1s. (End)
From _Gus Wiseman_, May 24 2024: (Start)
The terms together with their binary expansions and binary indices begin:
   0:      0 ~ {}
   1:      1 ~ {1}
   2:     10 ~ {2}
   3:     11 ~ {1,2}
   5:    101 ~ {1,3}
   6:    110 ~ {2,3}
   7:    111 ~ {1,2,3}
  11:   1011 ~ {1,2,4}
  13:   1101 ~ {1,3,4}
  14:   1110 ~ {2,3,4}
  15:   1111 ~ {1,2,3,4}
  23:  10111 ~ {1,2,3,5}
  27:  11011 ~ {1,2,4,5}
  29:  11101 ~ {1,3,4,5}
  30:  11110 ~ {2,3,4,5}
  31:  11111 ~ {1,2,3,4,5}
  47: 101111 ~ {1,2,3,4,6}
  55: 110111 ~ {1,2,3,5,6}
  59: 111011 ~ {1,2,4,5,6}
  61: 111101 ~ {1,3,4,5,6}
  62: 111110 ~ {2,3,4,5,6}
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Vladimir Shevelev, On the Basis Polynomials in the Theory of Permutations with Prescribed Up-Down Structure, arXiv:0801.0072 [math.CO], 2007-201. See Section 14.
Vladimir Shevelev, Binomial Coefficient Predictors, Journal of Integer Sequences, Vol. 14 (2011), Article 11.2.8.
Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS, 12 (2012), #A1.
Index entries for sequences related to binary expansion of n.

Cf. A000120, A007088, A011371, A014132, A023416, A023532, A029931.
Cf. A181741 (primes), union of A081118 and A000918, apart from initial -1.
Cf. A065442, A160502, A265705.
For least binary index (instead of rank) we have A001511.
Applying A019565 (Heinz number of binary indices) gives A077011.
For greatest binary index we have A029837 or A070939, opposite A070940.
Row minima of A118462 (binary ranks of strict partitions).
For sum instead of minimum we have A372888, non-strict A372890.
A000009 counts strict partitions, ranks A005117.
A048675 gives binary rank of prime indices, distinct A087207.
A048793 lists binary indices, product A096111, reverse A272020.
A277905 groups all positive integers by binary rank of prime indices.
Cf. A000041, A005940, A018819, A147655, A225620, A246868.

a089633 n = a089633_list !! (n-1)
a089633_list = [2 ^ t - 2 ^ k - 1 | t <- [1..], k <- [t-1,t-2..0]]
-- Reinhard Zumkeller, Feb 23 2012

seq(seq(2^a-1-2^b,b=a-1..0,-1),a=1..11); # Robert Israel, Dec 14 2018

fQ[n_] := DigitCount[n, 2, 0] < 2; Select[ Range[0, 2^10], fQ] (* Robert G. Wilson v, Aug 02 2012 *)

{insq(n) = local(dd, hf, v); v=binary(n);hf=length(v);dd=sum(i=1,hf,v[i]);if(dd<=hf-2,-1,1)}
{for(w=0,1536,if(insq(w)>=0,print1(w,", ")))}
\\ Douglas Latimer, May 07 2013

isoka(n) = #select(x->(x==0), binary(n)) <= 1; \\ Michel Marcus, Dec 14 2018

from itertools import count, islice
def A089633_gen(): # generator of terms
    return ((1<A089633_list = list(islice(A089633_gen(),30)) # Chai Wah Wu, Feb 10 2023

from math import isqrt, comb
def A089633(n): return (1<<(a:=(isqrt((n<<3)+1)-1>>1)+1))-(1<Chai Wah Wu, Dec 19 2024

A023416(a(n)) <= 1; A023416(a(n)) = A023532(n-2) for n>1;
A000120(a(u)) <= A000120(a(v)) for uA000120(a(n)) = A003056(n).
a(0)=0, n>0: a(n+1) = Min{m>n: BinOnes(a(n))<=BinOnes(m)} with BinOnes=A000120.
If m = floor((sqrt(8*n+1) - 1) / 2), then a(n) = 2^(m+1) - 2^(m*(m+3)/2 - n) - 1. - Carl R. White, Feb 10 2009
A029931(a(n)) = n and A029931(m) != n for m < a(n). - Reinhard Zumkeller, Feb 28 2014
A265705(a(n),k) = A265705(a(n),a(n)-k), k = 0 .. a(n). - Reinhard Zumkeller, Dec 15 2015
a(A014132(n)-1) = 2*a(n-1)+1 for n >= 1. - Robert Israel, Dec 14 2018
Sum_{n>=1} 1/a(n) = A065442 + A160502 = 3.069285887459... . - Amiram Eldar, Jan 09 2024
A019565(a(n)) = A077011(n). - Gus Wiseman, May 24 2024

A003817 a(0) = 0, a(n) = a(n-1) OR n.

Original entry on oeis.org

0, 1, 3, 3, 7, 7, 7, 7, 15, 15, 15, 15, 15, 15, 15, 15, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63

Offset: 0

Marc LeBrun

Also, 0+1+2+...+n in lunar arithmetic in base 2 written in base 10. - N. J. A. Sloane, Oct 02 2010

For n>0: replace all 0's with 1's in binary representation of n. - Reinhard Zumkeller, Jul 14 2003

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 [From _Reinhard Zumkeller_, Nov 14 2009]
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Reinhard Zumkeller, Logical Convolutions
Index entries for sequences related to dismal (or lunar) arithmetic

This is Guy Steele's sequence GS(6, 6) (see A135416).
Cf. A000004, A142149, A086099, A142150, A142151, A001477, A062383.
Cf. A167832, A167878. - Reinhard Zumkeller, Nov 14 2009
Cf. A179526; subsequence of A007448. - Reinhard Zumkeller, Jul 18 2010
Cf. A265705.

import Data.Bits ((.|.))
a003817 n = if n == 0 then 0 else 2 * a053644 n - 1
a003817_list = scanl (.|.) 0 [1..] :: [Integer]
-- Reinhard Zumkeller, Dec 08 2012, Jan 15 2012

A003817 := n -> n + Bits:-Nand(n, n):
seq(A003817(n), n=0..61); # Peter Luschny, Sep 23 2019

a[0] = 0; a[n_] := a[n] = BitOr[ a[n-1], n]; Table[a[n], {n, 0, 61}] (* Jean-François Alcover, Dec 19 2011 *)
nxt[{n_,a_}]:={n+1,BitOr[a,n+1]}; Transpose[NestList[nxt,{0,0},70]] [[2]] (* Harvey P. Dale, May 06 2016 *)
2^BitLength[Range[0,100]]-1 (* Paolo Xausa, Feb 08 2024 *)

a(n)=1<<(log(2*n+1)\log(2))-1 \\ Charles R Greathouse IV, Dec 08 2011

def a(n): return 0 if n==0 else 1 + 2*a(int(n/2)) # Indranil Ghosh, Apr 28 2017

def A003817(n): return (1<Chai Wah Wu, Jul 17 2024

a(n) = a(n-1) + n*(1-floor(a(n-1)/n)). If 2^(k-1) <= n < 2^k, a(n) = 2^k - 1. - Benoit Cloitre, Aug 25 2002
a(n) = 1 + 2*a(floor(n/2)) for n > 0. - Benoit Cloitre, Apr 04 2003
G.f.: (1/(1-x)) * Sum_{k>=0} 2^k*x^2^k. - Ralf Stephan, Apr 18 2003
a(n) = 2*A053644(n)-1 = A092323(n) + A053644(n). - Reinhard Zumkeller, Feb 15 2004; corrected by Anthony Browne, Jun 26 2016
a(n) = OR{k OR (n-k): 0<=k<=n}. - Reinhard Zumkeller, Jul 15 2008
For n>0: a(n+1) = A035327(n) + n = A035327(n) XOR n. - Reinhard Zumkeller, Nov 14 2009
A092323(n+1) = floor(a(n)/2). - Reinhard Zumkeller, Jul 18 2010
a(n) = A265705(n,0) = A265705(n,n). - Reinhard Zumkeller, Dec 15 2015
a(n) = A062383(n) - 1.
G.f. A(x) satisfies: A(x) = 2*A(x^2)*(1 + x) + x/(1 - x). - Ilya Gutkovskiy, Aug 31 2019
a(n) >= A175039(n) - Austin Shapiro, Dec 29 2022

A029578 The natural numbers interleaved with the even numbers.

Original entry on oeis.org

0, 0, 1, 2, 2, 4, 3, 6, 4, 8, 5, 10, 6, 12, 7, 14, 8, 16, 9, 18, 10, 20, 11, 22, 12, 24, 13, 26, 14, 28, 15, 30, 16, 32, 17, 34, 18, 36, 19, 38, 20, 40, 21, 42, 22, 44, 23, 46, 24, 48, 25, 50, 26, 52, 27, 54, 28, 56, 29, 58, 30, 60, 31, 62, 32, 64, 33, 66, 34, 68, 35, 70, 36, 72

Offset: 0

N. J. A. Sloane

a(n) = number of ordered, length two, compositions of n with at least one odd summand - Len Smiley, Nov 25 2001
Also number of 0's in n-th row of triangle in A071037. - Hans Havermann, May 26 2002
For n > 2: a(n) = number of odd terms in row n-2 of triangle A265705. - Reinhard Zumkeller, Dec 15 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A065423 (at least one even summand).

Cf. A001477, A005843, A009531, A071037, A211538 (partial sums), A265705.

import Data.List (transpose)
a029578 n =  (n - n `mod` 2) `div` (2 - n `mod` 2)
a029578_list = concat $ transpose [a001477_list, a005843_list]
-- Reinhard Zumkeller, Nov 27 2012

A029578:= func< n | (n + (n-2)*(n mod 2))/2 >;
[A029578(n): n in [0..80]]; // G. C. Greubel, Jan 22 2025

With[{nn=40},Riffle[Range[0,nn],Range[0,2nn,2]]] (* or *) LinearRecurrence[ {0,2,0,-1},{0,0,1,2},80] (* Harvey P. Dale, Aug 23 2015 *)

PARI
```
a(n)=if(n%2,n-1,n/2)
```

def A029578(n): return (n + (n-2)*(n%2))//2
print([A029578(n) for n in range(81)]) # G. C. Greubel, Jan 22 2025

a(n) = (3*n - 2 - (-1)^n*(n - 2))/4.
a(n+4) = 2*a(n+2) - a(n).
G.f.: x^2*(1 + 2*x)/(1-x^2)^2.
a(n) = floor((n+1)/2) + (n is odd)*floor((n+1)/2).
a(n) = (n - n mod 2)/(2 - n mod 2). - Reinhard Zumkeller, Jul 30 2002
a(n) = floor(n/2)*binomial(2, mod(n, 2)) - Paul Barry, May 25 2003
a(2*n) = n.
a(2*n-1) = 2*n-2.
a(-n) = -A065423(n+2).
a(n) = Sum_{k=0..floor((n-2)/2)} (C(n-k-2, k) mod 2)((1+(-1)^k)/2)*2^A000120(n-2k-2). - Paul Barry, Jan 06 2005
a(n) = Sum_{k=0..n-2} gcd(n-k-1, k+1). - Paul Barry, May 03 2005
For n>6: a(n) = floor(a(n-1)*a(n-2)/a(n-3)). - Reinhard Zumkeller, Mar 06 2011
E.g.f.: (1/4)*((x+2)*exp(-x) + (3*x-2)*exp(x)). - G. C. Greubel, Jan 22 2025

Explicated definition by Reinhard Zumkeller, Nov 27 2012

Title simplified by Sean A. Irvine, Feb 29 2020

A247648 Numbers whose binary expansion begins and ends with 1 and does not contain two adjacent zeros.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 21, 23, 27, 29, 31, 43, 45, 47, 53, 55, 59, 61, 63, 85, 87, 91, 93, 95, 107, 109, 111, 117, 119, 123, 125, 127, 171, 173, 175, 181, 183, 187, 189, 191, 213, 215, 219, 221, 223, 235, 237, 239, 245, 247, 251, 253

Offset: 1

N. J. A. Sloane, Sep 25 2014

Decimal equivalents of A247647.
A265716(a(n)) = A265705(2*a(n),a(n)) = 2*a(n). - Reinhard Zumkeller, Dec 15 2015
The viabin numbers of the integer partitions having distinct parts (for the definition of viabin number see comment in A290253). For example, 109 is in the sequence because it is the viabin number of the integer partition [5,4,2]; 121 is not in the sequence because it is the viabin number of the integer partition [5,4,4]. - Emeric Deutsch, Aug 29 2017

			109 is in the sequence because its binary expansion is 1101101.

Reinhard Zumkeller, Table of n, a(n) for n = 1..121392 (all terms < 2^24; first 1000 terms from Chai Wah Wu)
Andreas M. Hinz and Paul K. Stockmeyer, Discovering Fibonacci Numbers, Fibonacci Words, and a Fibonacci Fractal in the Tower of Hanoi, The Fibonacci Quarterly (2019) Vol. 57, No. 5, 72-83.
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Stefan Witzel, On panel-regular ~A2 lattices, Geom. Dedicata 191, 85-135 (2017).

Cf. A247647, A247649, A253085.
Cf. A247875 (complement).
Cf. A265716, A265705.

import Data.Set (singleton, deleteFindMin, insert)
a247648 n = a247648_list !! (n-1)
a247648_list = f $ singleton 1 where
   f s = x : f (insert (4 * x + 1) $ insert (2 * x + 1) s')
         where (x, s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 25 2014

vitopart := proc (n) local L, i, j, N, p, t: N := 2*n: L := ListTools:-Reverse(convert(N, base, 2)): j := 0: for i to nops(L) do if L[i] = 0 then j := j+1: p[j] := numboccur(L[1 .. i], 1) end if end do: sort([seq(p[t], t = 1 .. j)], `>=`) end proc: a := proc (n) if n = 1 then 1 elif `mod`(n, 2) = 0 then a((1/2)*n) elif `mod`(n, 2) = 1 and `mod`((1/2)*n-1/2, 2) = 0 then a((1/2)*n-1/2)+1 else a((1/2)*n-1/2) end if end proc: A := {}: for n to 254 do if a(n) = nops(vitopart(n)) then A := `union`(A, {n}) else end if end do: A; # program is based on my comment; the command vitopart(n) yields the integer partition having viabin number n. # Emeric Deutsch, Aug 29 2017

Select[Range@ 256, And[First@ # == Last@ # == 1, NoneTrue[Map[Length, Select[Split[#], First@ # == 0 &]], # > 1 &]] &@ IntegerDigits[#, 2] &] (* Michael De Vlieger, Aug 29 2017 *)

isok(k) = if (k%2, my(b=binary(k)); #select(x->(x==0), vector(#b-1, k, b[k]+b[k+1])) == 0); \\ Michel Marcus, Jun 15 2024

A247648_list = [n for n in range(1,10**5) if n % 2 and not '00' in bin(n)]
# Chai Wah Wu, Sep 25 2014

A080098 Triangle T(n,k) = n OR k, 0 <= k <= n, bitwise logical OR, read by rows.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 3, 3, 3, 4, 5, 6, 7, 4, 5, 5, 7, 7, 5, 5, 6, 7, 6, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 9, 11, 11, 13, 13, 15, 15, 9, 9, 10, 11, 10, 11, 14, 15, 14, 15, 10, 11, 10, 11, 11, 11, 11, 15, 15, 15, 15, 11, 11, 11, 11, 12, 13, 14, 15, 12, 13, 14, 15, 12, 13, 14, 15, 12

Offset: 0

Reinhard Zumkeller, Jan 28 2003

			Triangle begins:
   0,
   1,  1,
   2,  3,  2,
   3,  3,  3,  3,
   4,  5,  6,  7,  4,
   5,  5,  7,  7,  5,  5,
   6,  7,  6,  7,  6,  7,  6,
   7,  7,  7,  7,  7,  7,  7,  7,
   8,  9, 10, 11, 12, 13, 14, 15,  8,
   9,  9, 11, 11, 13, 13, 15, 15,  9,  9,
  10, 11, 10, 11, 14, 15, 14, 15, 10, 11, 10,
  ...

Rick L. Shepherd, Rows n = 0..500 of triangle, flattened
Eric Weisstein's World of Mathematics, OR.

Cf. A001316 (number of integers k such that T(n, k) = n in n-th row).
Cf. A350093 (row sums), A003986 (array).
Other triangles: A080099 (AND), A051933 (XOR), A265705 (IMPL), A102037 (CNIMPL).

import Data.Bits ((.|.))
a080098 n k = n .|. k :: Int
a080098_row n = map (a080098 n) [0..n]
a080098_tabl = map a080098_row [0..]
-- Reinhard Zumkeller, Aug 03 2014, Jul 05 2012

T[n_, k_] := n ~BitOr~ k;
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 01 2021 *)

def T(n, k): return n | k
print([T(n, k) for n in range(13) for k in range(n+1)]) # Michael S. Branicky, Dec 01 2021

A080099 Triangle T(n,k) = n AND k, 0<=k<=n, bitwise logical AND, read by rows.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 1, 2, 3, 0, 0, 0, 0, 4, 0, 1, 0, 1, 4, 5, 0, 0, 2, 2, 4, 4, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 1, 0, 1, 0, 1, 0, 1, 8, 9, 0, 0, 2, 2, 0, 0, 2, 2, 8, 8, 10, 0, 1, 2, 3, 0, 1, 2, 3, 8, 9, 10, 11, 0, 0, 0, 0, 4, 4, 4, 4, 8, 8, 8, 8, 12, 0, 1, 0, 1, 4, 5, 4, 5, 8, 9, 8, 9

Offset: 0

Reinhard Zumkeller, Jan 28 2003

A080100(n) = number of numbers k such that n AND k = 0 in n-th row of the triangular array.

			Triangle starts:
0
0 1
0 0 2
0 1 2 3
0 0 0 0 4
0 1 0 1 4 5
0 0 2 2 4 4 6
0 1 2 3 4 5 6 7
...

Rick L. Shepherd, Rows n=0..500 of triangle, flattened
Eric Weisstein's World of Mathematics, AND.

Cf. A080100, A222423 (row sums), A004198 (array).

Other triangles: A080098 (OR), A051933 (XOR), A265705 (IMPL), A102037 (CNIMPL).

import Data.Bits ((.&.))
a080099 n k = n .&. k :: Int
a080099_row n = map (a080099 n) [0..n]
a080099_tabl = map a080099_row [0..]
-- Reinhard Zumkeller, Aug 03 2014, Jul 05 2012

Column[Table[BitAnd[n, k], {n, 0, 15}, {k, 0, n}], Center] (* Alonso del Arte, Jun 19 2012 *)

T(n,k)=bitand(n,k) \\ Charles R Greathouse IV, Jan 26 2013

def T(n, k): return n & k
print([T(n, k) for n in range(14) for k in range(n+1)]) # Michael S. Branicky, Dec 16 2021

A051933 Triangle T(n,m) = Nim-sum (or XOR) of n and m, read by rows, 0<=m<=n.

Original entry on oeis.org

0, 1, 0, 2, 3, 0, 3, 2, 1, 0, 4, 5, 6, 7, 0, 5, 4, 7, 6, 1, 0, 6, 7, 4, 5, 2, 3, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 9, 10, 11, 12, 13, 14, 15, 0, 9, 8, 11, 10, 13, 12, 15, 14, 1, 0, 10, 11, 8, 9, 14, 15, 12, 13, 2, 3, 0, 11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 0

Offset: 0

N. J. A. Sloane, Dec 20 1999

			{0},
{1,0},
{2,3,0},
{3,2,1,0}, ...

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.
J. H. Conway, On Numbers and Games, Academic Press, p. 52.

Reinhard Zumkeller, Rows n = 0..127 of triangle, flattened first 50 rows by R. J. Mathar
Index entries for sequences related to Nim-sums

Cf. A224915 (row sums), A003987 (array), A051910 (Nim-product).

Other triangles: A080099 (AND), A080098 (OR), A265705 (IMPL), A102037 (CNIMPL), A002262 (k).

import Data.Bits (xor)
a051933 n k = n `xor` k :: Int
a051933_row n = map (a051933 n) [0..n]
a051933_tabl = map a051933_row [0..]
-- Reinhard Zumkeller, Aug 02 2014, Aug 13 2013

using IntegerSequences
A051933Row(n) = [Bits("XOR", n, k) for k in 0:n]
for n in 0:10 println(A051933Row(n)) end  # Peter Luschny, Sep 25 2021

nimsum := proc(a,b) local t1,t2,t3,t4,l; t1 := convert(a+2^20,base,2); t2 := convert(b+2^20,base,2); t3 := evalm(t1+t2); map(x->x mod 2, t3); t4 := convert(evalm(%),list); l := convert(t4,base,2,10); sum(l[k]*10^(k-1), k=1..nops(l)); end; # memo: adjust 2^20 to be much bigger than a and b
AT := array(0..N,0..N); for a from 0 to N do for b from a to N do AT[a,b] := nimsum(a,b); AT[b,a] := AT[a,b]; od: od:
# Alternative:
A051933 := (n, k) -> Bits:-Xor(n, k):
seq(seq(A051933(n, k), k=0..n), n=0..12); # Peter Luschny, Sep 23 2019

Flatten[Table[BitXor[m, n], {m, 0, 12}, {n, 0, m}]] (* Jean-François Alcover, Apr 29 2011 *)

More terms from Michael Lugo (mlugo(AT)thelabelguy.com), Dec 22 1999

A158582 Numbers with at least two zeros in their binary representation.

Original entry on oeis.org

4, 8, 9, 10, 12, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 60, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91

Offset: 1

Reinhard Zumkeller, Apr 16 2009

nonn

Subsequence of A158581; complement of A089633; A023416(a(n))>1.

A265705(a(n),k) != A265705(a(n),a(n)-k) for at least one k <= a(n). - Reinhard Zumkeller, Dec 15 2015

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007088, A023416.

Cf. A089633, A158581, A265705.

a158582 n = a158582_list !! (n-1)
a158582_list = [x | x <- [0..], a023416 x > 1]
-- Reinhard Zumkeller, Mar 31 2015

Select[Range[100],DigitCount[#,2,0]>1&] (* Harvey P. Dale, Jan 19 2015 *)

def A158582(n):
    def f(x):
        c = n+(((l:=(x+1).bit_length())+1)*(l-2)>>1)
        m = bin(x+1)[2:].find('0')
        c += m if m>-1 else l
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 24 2024

cp's OEIS Frontend

A265736 Row sums of triangle A265705.

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A053644 Most significant bit of n, msb(n); largest power of 2 less than or equal to n; write n in binary and change all but the first digit to zero.

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A089633 Numbers having no more than one 0 in their binary representation.

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A003817 a(0) = 0, a(n) = a(n-1) OR n.

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A029578 The natural numbers interleaved with the even numbers.

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A247648 Numbers whose binary expansion begins and ends with 1 and does not contain two adjacent zeros.

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