A352782 - OEIS

%I #19 Apr 04 2022 10:20:29

%S 0,1,2,4,8,3,16,32,64,5,6,12,128,9,256,512,1024,17,10,20,24,7,48,96,

%T 2048,33,18,36,4096,65,8192,16384,32768,129,34,68,40,11,80,160,192,13,

%U 14,28,384,25,768,1536,65536,257,66,132,72,19,144,288,131072,513

%N The binary expansion of a(n) encodes the runs of consecutive 1's in the binary expansion of n (see Comments section for precise definition).

%C For any nonnegative integer n:

%C - the binary expansion of n can be uniquely expressed as the concatenation of k = A069010(n) positive terms of A023758 separated by 0's:

%C       n = A023758(m_k+1) | 0 | A023758(m_{k-1}+1) | 0 | ... | 0 | A023758(m_1+1)

%C             (where | denotes binary concatenation)

%C - a(n) = ( Sum_{i = 1..k} 2^Sum_{j = 1..i} m_j ) / 2.

%C This sequence is a permutation of the nonnegative integers, with inverse A352783.

%H Rémy Sigrist, <a href="/A352782/b352782.txt">Table of n, a(n) for n = 0..8192</a>

%H Rémy Sigrist, <a href="/A352782/a352782.png">Colored logarithmic scatterplot of the first 2^20 terms</a> (where the color is function of A069010(n))

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F a(4*n+1) = 2*a(n)+1.

%F A000120(a(n)) = A069010(n).

%F a(A023758(k+1)) = 2^k for any k >= 0.

%F a(2^k) = A006125(k+1) for any k >= 0.

%F a(2^k-1) = A036442(k+1) for any k >= 0.

%F a(n) = n iff n = 0 or n belongs to A131865 or n/2 belongs to A131865.

%e For n = 89:

%e - the binary expansion of 89 is "1011001",

%e - "1011001" = "1" | 0 | "110" | 0 | "1"

%e             = A023758(1+1) | 0 | A023758(5+1) | 0 | A023758(1+1)

%e - so 2*a(89) = 2^(1+5+1) + 2^(5+1) + 2^1 = 194,

%e - and a(89) = 97.

%o (PARI) a(n) = { my (v=0, s=-1, z, o, i); while (n, n\=2^z=valuation(n,2); n\=2^o=valuation(n+1,2); n\=2; i=(o+z)*(o+z-1)/2 + o; v+=2^s+=i); v }

%Y Cf. A000120, A006125, A023758, A036442, A069010, A131865, A352783 (inverse).

%K nonn,base

%O 0,3

%A _Rémy Sigrist_, Apr 02 2022