A342410 The binary expansion of a(n) corresponds to that of n where all the 1's have been replaced by 0's except in the last run of 1's.

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

Offset: 0

Author

Rémy Sigrist, Apr 25 2021

Keywords

• nonn
• base
• easy

Comments

In other words, this sequence gives the last run of 1's in the binary expansion of a number.

Examples

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

Links

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

Crossrefs

Cf. A023758, A089309, A135481, A342126.

Programs

Mathematica

Array[FromDigits[If[Length[s=Split@IntegerDigits[#,2]]>1,Flatten[s[[-2;;]]],First@s],2]&,100,0] (* Giorgos Kalogeropoulos, Apr 27 2021 *)

PARI

a(n) = { if (n, my (z=valuation(n, 2), o=valuation(n/2^z+1, 2)); (2^o-1)*2^z, 0) }

Python

def A342410(n):
    if n == 0 : return 0
    for i, d in enumerate(bin(n)[2:].split('0')[::-1]):
        if d != '': return int(d+'0'*i,2) # Chai Wah Wu, Apr 29 2021

Formula

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

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

a(n) <= n with equality iff n belongs to A023758.