A372289 - OEIS

%I #19 Apr 27 2024 16:46:27

%S 1,5,3,21,5,13,7,85,9,21,11,53,13,29,15,341,17,37,19,85,21,45,23,213,

%T 25,53,27,117,29,61,31,1365,33,69,35,149,37,77,39,341,41,85,43,181,45,

%U 93,47,853,49,101,51,213,53,109,55,469,57,117,59,245,61,125,63,5461

%N a(n) = n*(2^e) + ((4^e)-1)/3, where e is the 2-adic valuation of n.

%C Construction: take the binary expansion of n, and substitute "01" for all trailing 0-bits that follow after its odd part (= A000265(n)). See the examples.

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

%F For n >= 0, a(2n+1) = 2n+1.

%e For n=4, "100" in binary, when we substitute 01's for the two trailing 0's, we obtain 21, "10101" in binary, therefore a(4) = 21.

%e For n=11, "1011" in binary, there are no trailing 0's, and thus no changes, therefore a(11) = 11.

%p a := proc(n) padic[ordp](n, 2): n*2^% + ((2^%)^2 - 1)/3 end:

%p seq(a(n), n = 1..64);  # _Peter Luschny_, Apr 27 2024

%t a[n_]:=n*(2^IntegerExponent[n, 2]) + ((4^IntegerExponent[n, 2]) - 1)/3; Array[a, 75] (* _Stefano Spezia_, Apr 26 2024 *)

%o (PARI) A372289(n) = { my(e=valuation(n,2)); ((n*(2^e)) + (((4^e)-1)/3)); };

%o (Python)

%o def A372289(n): return (n<<(e:=(~n & n-1).bit_length()))+((1<<(e<<1))-1)//3 # _Chai Wah Wu_, Apr 26 2024

%Y Cf. A000265, A005408 (odd bisection), A007814, A371094 [= a(3n+1)].

%K nonn

%O 1,2

%A _Antti Karttunen_ and _Ali Sada_, Apr 26 2024