A003817 -id:A003817 - cp's OEIS frontend

A142149 a(n) = XOR{k OR (n-k): 0<=k<=n}.

0, 1, 3, 3, 6, 5, 5, 7, 12, 9, 15, 11, 10, 13, 9, 15, 24, 17, 27, 19, 30, 21, 29, 23, 20, 25, 23, 27, 18, 29, 17, 31, 48, 33, 51, 35, 54, 37, 53, 39, 60, 41, 63, 43, 58, 45, 57, 47, 40, 49, 43, 51, 46, 53, 45, 55, 36, 57, 39, 59, 34, 61, 33, 63, 96, 65, 99, 67, 102, 69, 101, 71

Offset: 0

Reinhard Zumkeller, Jul 15 2008

a(n) = XOR{k AND (n-k): 0<=k<=n}.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Reinhard Zumkeller, Logical Convolutions

Cf. A003817, A000004, A086099, A142150, A142151, A001477.

import Data.Bits (xor, (.|.))
a142149 :: Integer -> Integer
a142149 = foldl xor 0 . zipWith (.|.) [0..] . reverse . enumFromTo 1
-- Reinhard Zumkeller, Mar 31 2015

a(n)=if(n%2, n, bitxor(n, n/2)) \\ Charles R Greathouse IV, Jul 01 2022

def A142149(n): return n if n&1 else (n^ n>>1) # Chai Wah Wu, Jun 29 2022

a(2*n) = A048724(n) and a(2*n+1) = A005408(n).

A167832 A167831(n) + n.

Original entry on oeis.org

0, 2, 4, 6, 8, 9, 9, 9, 9, 9, 20, 22, 24, 26, 28, 29, 29, 29, 29, 29, 40, 42, 44, 46, 48, 49, 49, 49, 49, 49, 60, 62, 64, 66, 68, 69, 69, 69, 69, 69, 80, 82, 84, 86, 88, 89, 89, 89, 89, 89, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99

Offset: 0

Reinhard Zumkeller, Nov 14 2009

No carry occurs when calculating a(n) by adding A167831(n) to n in decimal arithmetic.

R. Zumkeller, Table of n, a(n) for n = 0..9999

Cf. A167878, A003817 for the ternary and binary cases.

b167832 n = a167831 n + n  -- Reinhard Zumkeller, Mar 15 2014

A218388 Bitwise OR of all divisors of n.

Original entry on oeis.org

1, 3, 3, 7, 5, 7, 7, 15, 11, 15, 11, 15, 13, 15, 15, 31, 17, 31, 19, 31, 23, 31, 23, 31, 29, 31, 27, 31, 29, 31, 31, 63, 43, 51, 39, 63, 37, 55, 47, 63, 41, 63, 43, 63, 47, 63, 47, 63, 55, 63, 51, 63, 53, 63, 63, 63, 59, 63, 59, 63, 61, 63, 63, 127, 77, 127

Offset: 1

Reinhard Zumkeller, Oct 27 2012

a(n) <= A003817(n); a(A000040(n)) = A000040(n).

			n=20: divisors(20) = {1, 2, 4, 5, 10, 20}, 00001 OR 00010 OR 00100 OR 00101 OR 01010 OR 10100 = 11111 -> a(20) = 31;
n=21: divisors(21) = {1, 3, 7, 21}, 00001 OR 00011 OR 00111 OR 10101 = 10111 -> a(21) = 23;
n=22: divisors(22) = {1, 2, 11, 22}, 00001 OR 00010 OR 01011 OR 10110 = 11111 -> a(22) = 31;
n=23: divisors(23) = {1, 23}, 00001 OR 10111 = 10111 -> a(23) = 23;
n=24: divisors(24) = {1, 2, 3, 4, 6, 8, 12, 24}, 00001 OR 00010 OR 00011 OR 00100 OR 00110 OR 01000 OR 01100 OR 11000 = 11111 -> a(24) = 31;
n=25: divisors(25) = {1, 5, 25}, 00001 OR 00101 OR 11001 = 11101 -> a(25) = 29.

Reinhard Zumkeller, Table of n, a(n) for n = 1..8191
Eric Weisstein's World of Mathematics, OR
Wikipedia, Bitwise operation OR
Index entries for sequences related to binary expansion of n

Cf. A027750, A000225 (subsequence), A123345, A218403.

import Data.Bits ((.|.))
a218388 = foldl1 (.|.) . a027750_row :: Integer -> Integer

Table[BitOr@@Divisors[n],{n,70}] (* Harvey P. Dale, Feb 27 2013 *)

A361644 Irregular triangle T(n, k), n >= 0, k = 1..max(1, 2^(A005811(n)-1)), read by rows; the n-th row lists the integers with the same binary length as n and whose partial sums of run lengths are included in those of n.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 7, 4, 5, 6, 7, 6, 7, 7, 8, 15, 8, 9, 14, 15, 8, 9, 10, 11, 12, 13, 14, 15, 8, 11, 12, 15, 12, 15, 12, 13, 14, 15, 14, 15, 15, 16, 31, 16, 17, 30, 31, 16, 17, 18, 19, 28, 29, 30, 31, 16, 19, 28, 31, 16, 19, 20, 23, 24, 27, 28, 31

Offset: 0

Rémy Sigrist, Mar 19 2023

In other words, the n-th row contains the numbers k with the same binary length as n and for any i >= 0, if the i-th bit and the (i+1)-th bit in k are different then they are also different in n (i = 0 corresponding to the least significant bit).

The value m appears 2^A092339(m) times in the triangle (see A361674).

			Triangle begins (in decimal and in binary):
  n   n-th row      bin(n)  n-th row in binary
  --  ------------  ------  ------------------
   0  0                  0  0
   1  1                  1  1
   2  2, 3              10  10, 11
   3  3                 11  11
   4  4, 7             100  100, 111
   5  4, 5, 6, 7       101  100, 101, 110, 111
   6  6, 7             110  110, 111
   7  7                111  111
   8  8, 15           1000  1000, 1111
   9  8, 9, 14, 15    1001  1000, 1001, 1110, 1111
.
For n = 9:
- the binary expansion of 9 is "1001",
- the corresponding run lengths are 1, 2, 1,
- so the 9th row contains the values with the following run lengths:
      1, 2, 1  ->   9 ("1001" in binary)
      1,  2+1  ->   8 ("1000" in binary)
      1+2,  1  ->  14 ("1110" in binary)
       1+2+1   ->  15 ("1111" in binary)

Rémy Sigrist, Table of n, a(n) for n = 0..9841 (rows for n = 0..511 flattened)
Index entries for sequences related to binary expansion of n

Cf. A003817, A005811, A092339, A101211, A225081, A342126, A361645, A361646, A361674, A361676.

row(n) = { my (r = []); while (n, my (v = valuation(n+n%2, 2)); n \= 2^v; r = concat(v, r)); my (s = [if (#r, 2^r[1]-1, 0)]); for (k = 2, #r, s = concat(s * 2^r[k], [(h+1)*2^r[k]-1|h<-s]);); vecsort(s); }

T(n, 1) = A342126(n).

T(n, max(1, 2^(A005811(n)-1))) = A003817(n).

A092323 2^m - 1 appears 2^m times.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Feb 15 2004

Or, write n in binary and change the most significant bit to 0 and all other bits to 1.
a(n) = A053644(n) - 1 = A003817(n) - A053644(n).
a(n) = floor(A003817(n-1)/2). [Reinhard Zumkeller, Jul 18 2010]

Cf. A003817, A053644.

[n le 2 select n-1 else Self(Floor(n/2))*2+1: n in [1..100]]; // Vincenzo Librandi, Jun 27 2016

Table[FromDigits[#, 2] &@ Table[1, {IntegerLength[n, 2] - 1}], {n, 80}] (* Michael De Vlieger, Jun 26 2016 *)
Table[Table[2^m-1,2^m],{m,0,6}]//Flatten (* Harvey P. Dale, May 22 2021 *)
Table[PadRight[{},2^m,2^m-1],{m,0,6}]//Flatten (* Harvey P. Dale, Aug 18 2025 *)

a(n) = if n=1 then 0 else a(floor(n/2))*2 + 1.

a(1)=0, a(2n) = 2*a(n)+1, a(2n+1) = a(2n). - Ralf Stephan, Nov 18 2010

A167878 A167877(n) + n.

Original entry on oeis.org

0, 2, 2, 6, 8, 8, 8, 8, 8, 18, 20, 20, 24, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 54, 56, 56, 60, 62, 62, 62, 62, 62, 72, 74, 74, 78, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80

Offset: 0

Reinhard Zumkeller, Nov 14 2009

No carry occurs when calculating a(n) by adding A167877(n) to n in ternary arithmetic.

R. Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A007089, see A167832, A003817 for the decimal and binary cases.

a167878 n = a167877 n + n  -- Reinhard Zumkeller, Mar 15 2014

A340632 a(n) in binary is a run of 1-bits from the most significant 1-bit of n down to the least significant 1-bit of n, inclusive.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 7, 8, 15, 14, 15, 12, 15, 14, 15, 16, 31, 30, 31, 28, 31, 30, 31, 24, 31, 30, 31, 28, 31, 30, 31, 32, 63, 62, 63, 60, 63, 62, 63, 56, 63, 62, 63, 60, 63, 62, 63, 48, 63, 62, 63, 60, 63, 62, 63, 56, 63, 62, 63, 60, 63, 62, 63, 64, 127, 126

Offset: 0

Kevin Ryde, Jan 13 2021

			n    = 172 = binary 10101100;
a(n) = 252 = binary 11111100.

Kevin Ryde, Table of n, a(n) for n = 0..8192

Cf. A023758 (distinct terms).

Cf. A006519, A062383, A142151.

PARI
```
a(n) = if(n, 2<
				
```
Python
```
def a(n): return (1<
				
```

a(n) = A062383(n) - A006519(n) for n>=1.
a(n) = A003817(n) - A135481(n-1).
a(n) = n + A334045(n) (filling in 0-bits, including n=0 by taking A334045(0)=0).
a(n) = A142151(n-1) + 1.
G.f.: x/(1-x) + Sum_{k>=0} 2^k*x^(2^k)*(1/(1-x) - 1/(1-x^(2^(k+1)))).

A342698 For any number n with binary expansion (b(1), b(2), ..., b(k)), the binary expansion of a(n) is (floor((b(k)+b(1)+b(2))/2), floor((b(1)+b(2)+b(3))/2), ..., floor((b(k-1)+b(k)+b(1))/2)).

Original entry on oeis.org

0, 1, 1, 3, 0, 7, 7, 7, 0, 9, 5, 15, 12, 15, 15, 15, 0, 17, 1, 19, 8, 27, 15, 31, 24, 25, 29, 31, 28, 31, 31, 31, 0, 33, 1, 35, 0, 35, 7, 39, 16, 49, 21, 55, 28, 63, 31, 63, 48, 49, 49, 51, 56, 59, 63, 63, 56, 57, 61, 63, 60, 63, 63, 63, 0, 65, 1, 67, 0, 67, 7

Offset: 0

Rémy Sigrist, Mar 18 2021

This sequence is a variant of A342697; here we deal with bit triples in a "cyclic" binary representation of n.

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     1      10          1
   3     3      11         11
   4     0     100          0
   5     7     101        111
   6     7     110        111
   7     7     111        111
   8     0    1000          0
   9     9    1001       1001
  10     5    1010        101
  11    15    1011       1111
  12    12    1100       1100
  13    15    1101       1111
  14    15    1110       1111
  15    15    1111       1111

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A003817, A342697, A342699 (fixed points), A342700.

a(n) = my (w=#binary(n)); sum(k=0, w-1, ((bittest(n, (k-1)%w)+bittest(n, k%w)+bittest(n, (k+1)%w))>=2) * 2^k)

a(n) + A342700(n) = A003817(n).

a(n) = n iff n belongs to A342699.

A355222 The k-th leftmost digit of a(n) is the greatest of the k leftmost digits of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 22, 22, 23, 24, 25, 26, 27, 28, 29, 33, 33, 33, 33, 34, 35, 36, 37, 38, 39, 44, 44, 44, 44, 44, 45, 46, 47, 48, 49, 55, 55, 55, 55, 55, 55, 56, 57, 58, 59, 66, 66, 66, 66, 66, 66, 66, 67

Offset: 0

Rémy Sigrist, Jun 24 2022

Leading zeros are ignored.

			For n = 1402: max({1}) = 1, max({1, 4}) = 4, max({1, 4, 0}) = 4, max({1, 4, 0, 2}) = 4, so a(1402) = 1444.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

See A355221, A355223 and A355224 for similar sequences.

Cf. A003817 (binary analog), A009994 (fixed points).

a(n, base=10) = { my (d=digits(n, base), m=-oo); for (k=1, #d, d[k]=m=max(m, d[k])); fromdigits(d, base) }

def a(n):
    s, m = str(n), "0"
    return int("".join((m:=max(m, s[k])) for k in range(len(s))))
print([a(n) for n in range(68)]) # Michael S. Branicky, Jun 24 2022

from itertools import accumulate
def A355222(n): return int(''.join(accumulate(str(n),func=max))) # Chai Wah Wu, Jun 25 2022

a(n) >= n with equality iff n belongs to A009994.

a(a(n)) = a(n).

A179526 (3^k - 1)/2 appears 3^(k-1) times, k>0.

Original entry on oeis.org

1, 4, 4, 4, 13, 13, 13, 13, 13, 13, 13, 13, 13, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121

Offset: 0

Reinhard Zumkeller, Jul 18 2010

nonn

Cf. A003462, A007448, A003817.

Table[PadRight[{},3^(k-1),(3^k-1)/2],{k,5}]//Flatten (* Harvey P. Dale, May 30 2021 *)

a(n+1) = 3*a(floor(n/3)) + 1; a(0) = 1.

Erroneous comment deleted by Reinhard Zumkeller, Jul 23 2010

cp's OEIS Frontend

A142149 a(n) = XOR{k OR (n-k): 0<=k<=n}.

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A167832 A167831(n) + n.

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A218388 Bitwise OR of all divisors of n.

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Mathematica

A361644 Irregular triangle T(n, k), n >= 0, k = 1..max(1, 2^(A005811(n)-1)), read by rows; the n-th row lists the integers with the same binary length as n and whose partial sums of run lengths are included in those of n.

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A092323 2^m - 1 appears 2^m times.

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Magma

Mathematica

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A167878 A167877(n) + n.

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Haskell

A340632 a(n) in binary is a run of 1-bits from the most significant 1-bit of n down to the least significant 1-bit of n, inclusive.

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A342698 For any number n with binary expansion (b(1), b(2), ..., b(k)), the binary expansion of a(n) is (floor((b(k)+b(1)+b(2))/2), floor((b(1)+b(2)+b(3))/2), ..., floor((b(k-1)+b(k)+b(1))/2)).

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