A341746 If the runs in the binary expansion of n are (r_1, ..., r_k), then the runs in the binary expansion of a(n) are (r_1 + ... + r_k, r_1, ..., r_{k-1}).

1, 6, 3, 14, 29, 28, 7, 30, 123, 122, 61, 60, 121, 120, 15, 62, 503, 502, 251, 250, 501, 500, 125, 124, 499, 498, 249, 248, 497, 496, 31, 126, 2031, 2030, 1015, 1014, 2029, 2028, 507, 506, 2027, 2026, 1013, 1012, 2025, 2024, 253, 252, 2023, 2022, 1011, 1010

Offset: 1

Author

Rémy Sigrist, Feb 18 2021

Keywords

• nonn
• base

Comments

This sequence is related to A341694 (see Formula section).

All terms are distinct.

If a(n) > n, then a(n) does not appear in A341699.

Examples

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   1     1       1          1
   2     6      10        110
   3     3      11         11
   4    14     100       1110
   5    29     101      11101
   6    28     110      11100
   7     7     111        111
   8    30    1000      11110
   9   123    1001    1111011
  10   122    1010    1111010
  11    61    1011     111101
  12    60    1100     111100
  13   121    1101    1111001
  14   120    1110    1111000
  15    15    1111       1111
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..8192
• Rémy Sigrist, Binary plot of the first 255 terms
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A005811, A070939, A090996, A126646, A341694, A341699.

Programs

PARI

toruns(n) = { my (r=[]); while (n, my (v=valuation(n+n%2, 2)); n\=2^v; r=concat(v, r)); r }
fromruns(r) = { my (v=0); for (k=1, #r, v=(v+k%2)*2^r[k]-k%2); v }
a(n) = { my (r=toruns(n)); fromruns(concat(vecsum(r), r[1..#r-1])) }

Formula

A341694(a(n), k) = A341694(n, k+1).

a(n) = n iff n belongs to A126646.

A090996(a(n)) = A070939(n).

A090996(a(n)) > A070939(a(n)) / 2.

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