A371442 For any positive integer n with binary digits (b_1, ..., b_w) (where b_1 = 1), the binary digits of a(n) are (b_1, b_3, ..., b_{2*ceiling(w/2)-1}); a(0) = 0.

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

Offset: 0

Author

Rémy Sigrist, Mar 24 2024

Keywords

• nonn
• base
• easy

Comments

In other words, we keep odd-indexed bits.

For any v > 0, the value v appears A003945(A070939(v)) times in the sequence.

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     1      10          1
   3     1      11          1
   4     2     100         10
   5     3     101         11
   6     2     110         10
   7     3     111         11
   8     2    1000         10
   9     2    1001         10
  10     3    1010         11
  11     3    1011         11
  12     2    1100         10
  13     2    1101         10
  14     3    1110         11
  15     3    1111         11
		

Links

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

Crossrefs

See A371459 for the sequence related to even-indexed bits.

See A059905 and A063694 for similar sequences.

Cf. A000695, A001196, A003945, A070939, A165199, A371461.

Programs

Mathematica

A371442[n_] := FromDigits[IntegerDigits[n, 2][[1;;-1;;2]], 2];
Array[A371442, 100, 0] (* Paolo Xausa, Mar 28 2024 *)

PARI

a(n) = { my (b = binary(n)); fromdigits(vector(ceil(#b/2), k, b[2*k-1]), 2); }

Python

def a(n): return int(bin(n)[::2], 2)

Formula

a(A000695(n)) = n.

a(A001196(n)) = n.

a(A165199(n)) = a(n).