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, 11121, 111111, 11112, 1122, 11211, 1131, 114, 24, 231, 2211, 222, 2112

Offset

1,2

Comments

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

Links

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

Example

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
...

Mathematica

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

Prog

(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

Crossrefs

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

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

Sequence in context: A330521 A338191 A360227 * A267320 A303785 A262369

Adjacent sequences: A318924 A318925 A318926 * A318928 A318929 A318930

Keyword

nonn,base

Author

N. J. A. Sloane, Sep 09 2018

Status

approved