A328282 a(n) is the least k such that A175930(k) = n.

1, 3, 2, 15, 6, 4, 5, 255, 30, 12, 13, 16, 9, 11, 10, 65535, 510, 60, 61, 48, 25, 27, 26, 256, 33, 19, 18, 47, 22, 20, 21, 4294967295, 131070, 1020, 1021, 240, 121, 123, 122, 768, 97, 51, 50, 111, 54, 52, 53, 65536, 513, 67, 66, 79, 38, 36, 37, 767, 94, 44, 45

Offset: 1

Author

Rémy Sigrist, Oct 11 2019

Keywords

• nonn
• base

Comments

To compute a(n):

- the binary representation of n has k = A000120(n) one bits,

- the binary representation of a(n) has k runs of consecutive equal bits,

- the length of the i-th run in a(n) has length 2^z where z is the number of zeros immediately following the i-th one bit in the binary representation of n,

- this division into sections starting with ones in n or corresponding to a run in a(n) is materialized by slashes in the example section.

Examples

			The first terms, alongside the binary representation of n and of a(n) with peer sections separated by slashes, are:
    n   a(n)   bin(n)   bin(a(n))
    --  -----  -------  ----------------
     1      1        1                 1
     2      3       10                11
     3      2      1/1               1/0
     4     15      100              1111
     5      6     10/1              11/0
     6      4     1/10              1/00
     7      5    1/1/1             1/0/1
     8    255     1000          11111111
     9     30    100/1            1111/0
    10     12    10/10             11/00
    11     13   10/1/1            11/0/1
    12     16    1/100            1/0000
    13      9   1/10/1            1/00/1
    14     11   1/1/10            1/0/11
    15     10  1/1/1/1           1/0/1/0
    16  65535    10000  1111111111111111
		

Crossrefs

Cf. A000120, A005811, A175930.

Programs

PARI

a(n)={ my (r=[], l, v=0); while (n, r=concat(l=1+valuation(n,2), r); n \= 2^l); for (i=1, #r, v *= 2^2^(r[i]-1); if (i%2, v += 2^2^(r[i]-1)-1)); v }

Formula

a(n) <= 2^n-1 with equality iff n is a power of 2.

A005811(a(n)) = A000120(n).