A136480 Number of trailing equal digits in binary representation of n.

1, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1, 4, 4, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1, 5, 5, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1, 4, 4, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1, 6, 6, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1, 4, 4, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1, 5, 5, 1, 1, 2, 2, 1, 1, 3, 3

Offset: 0

Author

Reinhard Zumkeller, Dec 31 2007

Keywords

• nonn
• base
• easy

Comments

a(even) = number of trailing binary zeros;

a(odd) = number of trailing binary ones.

For n>0, power of 2 associated with n^2 + n, e.g. n=4 gives 20, so a(4)=2. - Jon Perry, Sep 12 2014

Links

• James Spahlinger, Table of n, a(n) for n = 0..10000
• Francis Laclé, 2-adic parity explorations of the 3n+ 1 problem, hal-03201180v2 [cs.DM], 2021.
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A007814, A050603, A094267, A163575, A001511, A039963 (parity).

Programs

Haskell

a136480 0 = 1
a136480 n = a007814 $ n + mod n 2  -- Reinhard Zumkeller, Jul 22 2014

JavaScript

for (n=1;n<120;n++) {
m=n*n+n;
c=0;
while (m%2==0) {m/=2;c++;}
document.write(c+", ");
}  // Jon Perry, Sep 12 2014

Maple

A136480 := proc(n)
    if n = 0 then
        1;
    else
        A007814(n*(n+1)) ;
    end if;
end proc:
seq( A136480(n),n=0..80) ; # R. J. Mathar, Mar 20 2023

Mathematica

Length[Last[Split[IntegerDigits[#,2]]]]&/@Range[0,140]  (* Harvey P. Dale, Mar 31 2011 *)

PARI

a(n)=if (n, valuation(n+n%2,2), 1) \\ Charles R Greathouse IV, Oct 14 2013

Python

def A136480(n): return (~(m:=n+(n&1))& m-1).bit_length() # Chai Wah Wu, Jul 08 2022

Formula

a(n) = A050603(n-1) for n>0;

a(2*n + n mod 2) = a(n) + 1.

For n>0: a(n) = A007814(n + n mod 2).

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2. - Amiram Eldar, Sep 15 2022

a(n) = A007814(A002378(n)), n>0. - R. J. Mathar, Mar 20 2023