A374355 a(n) is the least fibbinary number f <= n such that n - f is also a fibbinary number whose binary expansion has no common 1's with that of f (where fibbinary numbers correspond to A003714).

0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 1, 4, 4, 4, 5, 0, 0, 0, 1, 0, 0, 2, 2, 8, 8, 8, 9, 8, 8, 10, 10, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 1, 4, 4, 4, 5, 16, 16, 16, 17, 16, 16, 18, 18, 16, 16, 16, 17, 20, 20, 20, 21, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 1, 4, 4, 4, 5, 0

Offset: 0

Author

Rémy Sigrist, Jul 06 2024

Keywords

• nonn
• base

Comments

To compute a(n): replace every other bit with zero (starting with the first bit) in each run of consecutive 1's in the binary expansion of n.

Examples

			The first terms, in binary and in decimal, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     0       1          0
   2     0      10          0
   3     1      11          1
   4     0     100          0
   5     0     101          0
   6     2     110         10
   7     2     111         10
   8     0    1000          0
   9     0    1001          0
  10     0    1010          0
  11     1    1011          1
  12     4    1100        100
  13     4    1101        100
  14     4    1110        100
  15     5    1111        101
  16     0   10000          0
		

Links

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

Crossrefs

Cf. A003714, A374354, A374356.

Programs

PARI

a(n) = { my (v = 0, e, x, y, b); while (n, x = y = 0; e = valuation(n, 2); for (k = 0, oo, if (bittest(n, e+k), n -= b = 2^(e+k); [x, y] = [y + b, x], v += y; break;););); return (v); }

Formula

a(n) = A374354(n, 0).

a(n) = n - A374356(n).

a(n) >= 0 with equality iff n is a fibbinary number.