cp's OEIS Frontend

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

%I A318927 #22 Aug 21 2025 08:45:07
%S A318927 1,11,2,12,111,21,3,13,121,1111,112,22,211,31,4,14,131,1211,122,1112,
%T A318927 11111,1121,113,23,221,2111,212,32,311,41,5,15,141,1311,132,1212,
%U A318927 12111,1221,123,1113
%N A318927 Take the binary expansion of n, starting with the most significant bit, and concatenate the lengths of the runs.
%C A318927 Obviously this compressed notation is useful only for n < 1023. A101211 is a version which works for all n.
%H A318927 Rémy Sigrist, <a href="/A318927/b318927.txt">Table of n, a(n) for n = 1..1022</a>
%e A318927 n, binary, run lengths, -> a(n)
%e A318927 1, [1], [1] -> 1
%e A318927 2, [1, 0], [1, 1] -> 11
%e A318927 3, [1, 1], [2] -> 2
%e A318927 4, [1, 0, 0], [1, 2] -> 12
%e A318927 5, [1, 0, 1], [1, 1, 1] -> 111
%e A318927 6, [1, 1, 0], [2, 1] -> 21
%e A318927 7, [1, 1, 1], [3] -> 3
%e A318927 ...
%t A318927 Array[FromDigits@ Flatten[IntegerDigits@ Length[#] & /@ Split@ IntegerDigits[#, 2]] &, 40] (* _Michael De Vlieger_, Feb 17 2022 *)
%o A318927 (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
%o A318927 (Python)
%o A318927 from itertools import groupby
%o A318927 def A318927(n): return int(''.join(str(len(list(g))) for k, g in groupby(bin(n)[2:]))) # _Chai Wah Wu_, Mar 11 2022
%Y A318927 Cf. A101211 (without concatenation, as rows), A227736 (rows reversed), A318926 (reverse concatenation).
%Y A318927 Cf. A382255 (Heinz numbers instead of concatenation of the run lengths).
%K A318927 nonn,base
%O A318927 1,2
%A A318927 _N. J. A. Sloane_, Sep 09 2018