A318927 Take the binary expansion of n, starting with the most significant bit, and concatenate the lengths of the runs.

1, 11, 2, 12, 111, 21, 3, 13, 121, 1111, 112, 22, 211, 31, 4, 14, 131, 1211, 122, 1112, 11111, 1121, 113, 23, 221, 2111, 212, 32, 311, 41, 5, 15, 141, 1311, 132, 1212, 12111, 1221, 123, 1113

Offset: 1

Author

N. J. A. Sloane, Sep 09 2018

Keywords

• nonn
• base

Comments

Obviously this compressed notation is useful only for n < 1023. A101211 is a version which works for all n.

Examples

			n, binary, run lengths, -> a(n)
1, [1], [1] -> 1
2, [1, 0], [1, 1] -> 11
3, [1, 1], [2] -> 2
4, [1, 0, 0], [1, 2] -> 12
5, [1, 0, 1], [1, 1, 1] -> 111
6, [1, 1, 0], [2, 1] -> 21
7, [1, 1, 1], [3] -> 3
...
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..1022

Crossrefs

Cf. A101211 (without concatenation, as rows), A227736 (rows reversed), A318926 (reverse concatenation).

Cf. A382255 (Heinz numbers instead of concatenation of the run lengths).

Programs

Mathematica

Array[FromDigits@ Flatten[IntegerDigits@ Length[#] & /@ Split@ IntegerDigits[#, 2]] &, 40] (* Michael De Vlieger, Feb 17 2022 *)

PARI

a(n) = { my(d=[], r); while(n, n>>=r=valuation(n+n%2, 2); d=concat(digits(r), d)); fromdigits(d) } \\ Rémy Sigrist, Feb 17 2022, edited by M. F. Hasler, Mar 11 2025

Python

from itertools import groupby
def A318927(n): return int(''.join(str(len(list(g))) for k, g in groupby(bin(n)[2:]))) # Chai Wah Wu, Mar 11 2022