cp's OEIS Frontend

A352782 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).

%I A352782 #19 Apr 04 2022 10:20:29
%S A352782 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 A352782 2048,33,18,36,4096,65,8192,16384,32768,129,34,68,40,11,80,160,192,13,
%U A352782 14,28,384,25,768,1536,65536,257,66,132,72,19,144,288,131072,513
%N A352782 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 A352782 For any nonnegative integer n:
%C A352782 - 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 A352782       n = A023758(m_k+1) | 0 | A023758(m_{k-1}+1) | 0 | ... | 0 | A023758(m_1+1)
%C A352782             (where | denotes binary concatenation)
%C A352782 - a(n) = ( Sum_{i = 1..k} 2^Sum_{j = 1..i} m_j ) / 2.
%C A352782 This sequence is a permutation of the nonnegative integers, with inverse A352783.
%H A352782 Rémy Sigrist, <a href="/A352782/b352782.txt">Table of n, a(n) for n = 0..8192</a>
%H A352782 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 A352782 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A352782 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A352782 a(4*n+1) = 2*a(n)+1.
%F A352782 A000120(a(n)) = A069010(n).
%F A352782 a(A023758(k+1)) = 2^k for any k >= 0.
%F A352782 a(2^k) = A006125(k+1) for any k >= 0.
%F A352782 a(2^k-1) = A036442(k+1) for any k >= 0.
%F A352782 a(n) = n iff n = 0 or n belongs to A131865 or n/2 belongs to A131865.
%e A352782 For n = 89:
%e A352782 - the binary expansion of 89 is "1011001",
%e A352782 - "1011001" = "1" | 0 | "110" | 0 | "1"
%e A352782             = A023758(1+1) | 0 | A023758(5+1) | 0 | A023758(1+1)
%e A352782 - so 2*a(89) = 2^(1+5+1) + 2^(5+1) + 2^1 = 194,
%e A352782 - and a(89) = 97.
%o A352782 (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 A352782 Cf. A000120, A006125, A023758, A036442, A069010, A131865, A352783 (inverse).
%K A352782 nonn,base
%O A352782 0,3
%A A352782 _Rémy Sigrist_, Apr 02 2022