A344220 -id:A344220 - cp's OEIS frontend

A344259 For any number n with binary expansion (b(1), ..., b(k)), the binary expansion of a(n) is (b(1), ..., b(ceiling(k/2))).

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

Offset: 0

Rémy Sigrist, May 13 2021

Leading zeros are ignored.

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

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

Cf. A006995, A020330, A162751, A344220.

Array[FromDigits[First@Partition[l=IntegerDigits[#,2],Ceiling[Length@l/2]],2]&,100,0] (* Giorgos Kalogeropoulos, May 14 2021 *)

PARI
```
a(n) = n\2^(#binary(n)\2)
```

def a(n): b = bin(n)[2:]; return int(b[:(len(b)+1)//2], 2)
print([a(n) for n in range(85)]) # Michael S. Branicky, May 14 2021

a(A020330(n)) = n.
a(A006995(n+1)) = A162751(n).
a(n XOR A344220(n)) = a(n) (where XOR denotes the bitwise XOR operator).

A370427 a(n) is the least k >= 0 such that n OR k is a binary palindrome (where OR denotes the bitwise OR operator).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 5, 4, 3, 2, 1, 0, 1, 0, 9, 8, 1, 0, 9, 8, 3, 2, 1, 0, 3, 2, 1, 0, 1, 0, 17, 16, 9, 8, 25, 24, 5, 4, 21, 20, 1, 0, 17, 16, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 0, 33, 32, 17, 16, 49, 48, 1, 0, 33, 32, 17, 16, 49, 48

Offset: 0

Rémy Sigrist, Feb 18 2024

The binary expansions of n and a(n) have no common 1's.

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

Cf. A006995, A030101, A175297, A344220 (XOR variant).

A370427[n_] := With[{r = IntegerReverse[n, 2]}, r - BitAnd[n, r]];
Array[A370427, 2^7, 0] (* Paolo Xausa, Feb 20 2024 *)

a(n) = my (r = fromdigits(Vecrev(binary(n)), 2)); r - bitand(n, r)

n AND a(n) = 0 (where AND denotes the bitwise AND operator).
a(n) = A030101(n) - (n AND A030101(n)).
a(n) = A030101(n) - A175297(n) (for any n > 0).
a(n) = 0 iff n belongs to A006995.

cp's OEIS Frontend

A344259 For any number n with binary expansion (b(1), ..., b(k)), the binary expansion of a(n) is (b(1), ..., b(ceiling(k/2))).

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