A377414 a(n) is the largest term of A126684, say b, such that n AND b = b (where AND denotes the bitwise AND operator).

0, 1, 2, 2, 4, 5, 4, 5, 8, 8, 10, 10, 8, 8, 10, 10, 16, 17, 16, 17, 20, 21, 20, 21, 16, 17, 16, 17, 20, 21, 20, 21, 32, 32, 34, 34, 32, 32, 34, 34, 40, 40, 42, 42, 40, 40, 42, 42, 32, 32, 34, 34, 32, 32, 34, 34, 40, 40, 42, 42, 40, 40, 42, 42, 64, 65, 64, 65

Offset: 0

Author

Rémy Sigrist, Oct 27 2024

Keywords

• nonn
• base
• easy

Comments

For any n > 0 with binary expansion (b_1 = 1, b_2, ..., b_k), the binary expansion of a(n) is (c_1, ..., c_k) where c_i = b_i when i is odd, c_i = 0 when i is even.

For any n, the value c = n - a(n) also belongs to A126684 and satisfies n AND c = c (see A377415).

Examples

			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     2      10         10
   3     2      11         10
   4     4     100        100
   5     5     101        101
   6     4     110        100
   7     5     111        101
   8     8    1000       1000
   9     8    1001       1000
  10    10    1010       1010
  11    10    1011       1010
  12     8    1100       1000
  13     8    1101       1000
  14    10    1110       1010
  15    10    1111       1010
		

Links

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

Crossrefs

See A063694, A063695 and A374356 for similar sequences.

Cf. A000975, A070939, A126684, A371442, A377415.

Programs

PARI

a(n) = { my (v = 0, x = exponent(n), y); while (n, n -= 2^y = exponent(n); if (x%2 == y%2, v += 2^y;);); return (v); }

Formula

a(n) <= n with equality iff n belongs to A126684.

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

a(2*n) = 2*a(n).

a(n) = n AND A000975(A070939(n)). - Alan Michael Gómez Calderón, Jun 27 2025