A175930 - OEIS

%I #13 Oct 02 2021 11:06:27

%S 1,3,2,6,7,5,3,7,13,15,14,10,11,7,4,12,15,27,26,30,31,29,15,11,21,23,

%T 22,14,15,9,5,13,25,31,30,54,55,53,27,31,61,63,62,58,59,31,28,20,23,

%U 43,42,46,47,45,23,15,29,31,30,18,19,11,6,14,27,51,50,62,63,61,31,55,109,111

%N Concatenation of run lengths in binary expansion of n, written in base 2, then converted to base 10.

%H Rémy Sigrist, <a href="/A175930/b175930.txt">Table of n, a(n) for n = 1..8192</a>

%F From _Rémy Sigrist_, Jul 02 2019: (Begin)

%F a(2^k-1) = k for any k > 0.

%F a(2^k) = A004755(k) for any k > 0.

%F (End)

%e 6 = 110, two runs, lengths 2 and 1, so we write down 101 and convert it to base 10, getting 5. So a(6) = 5.

%t f[n_] := FromDigits[Flatten[IntegerDigits[Length /@ Split[IntegerDigits[n, \ 2]], 2]], 2]

%t Array[f, NUMBER OF TERMS]

%o (PARI) a(n) = my (b=[]); while (n, my (x=valuation(n+(n%2), 2)); b = concat(binary(x), b); n \= 2^x); fromdigits(b, 2) \\ _Rémy Sigrist_, Jul 02 2019

%o (Python)

%o from itertools import groupby

%o def a(n):

%o     c = "".join(bin(len(list(g)))[2:] for k, g in groupby(bin(n)[2:]))

%o     return int(c, 2)

%o print([a(n) for n in range(1, 75)]) # _Michael S. Branicky_, Oct 02 2021

%Y Cf. A004755, A101211, A321226.

%K base,easy,look,nonn

%O 1,2

%A _Dylan Hamilton_, Oct 23 2010

%E Edited by _N. J. A. Sloane_, Oct 23 2010