A050603 A001511 with every term repeated.

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, 1, 1, 2, 2, 1, 1, 4

Offset: 0

Author

Antti Karttunen Jun 22 1999

Keywords

• nonn
• easy

Comments

Column 2 of A050600: a(n) = add1c(n,2).

Absolute values of A094267.

Consider the Collatz (or 3x+1) problem and the iterative sequence c(k) where c(0)=n is a positive integer and c(k+1)=c(k)/2 if c(k) is even, c(k+1)=(3*c(k)+1)/2 if c(k) is odd. Then a(n) is the minimum number of iterations in order to have c(a(n)) odd if n is even or c(a(n)) even if n is odd. - Benoit Cloitre, Nov 16 2001

Links

• James Spahlinger, Table of n, a(n) for n = 0..10000
• Cristian Cobeli, Mihai Prunescu, and Alexandru Zaharescu, A growth model based on the arithmetic Z-game, arXiv:1511.04315 [math.NT], 2015.
• Francis Laclé, 2-adic parity explorations of the 3n+1 problem, hal-03201180v2 [cs.DM], 2021.

Crossrefs

Bisection gives column 1 of A050600: A001511.

Cf. A007814, A053398, A094267, A136480.

Programs

Mathematica

With[{c=Table[Position[Reverse[IntegerDigits[n,2]],1,1,1],{n,110}]// Flatten}, Riffle[c,c]] (* Harvey P. Dale, Dec 06 2018 *)
a[n_] := IntegerExponent[Floor[n/2]+1, 2] + 1; Array[a, 100, 0] (* Amiram Eldar, May 22 2025 *)

PARI

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

PARI

{a(n) = my(A); if( n<0, 0, A = sum(k=1, length( binary(n+2)) - 1, x^(2^k) / (1 - x^(2^k)), x^3 * O(x^n)); polcoeff( A * (1 + x) / x^2, n))}; /* Michael Somos, May 11 2014 */

Python

def A050603(n): return ((m:=n>>1)&~(m+1)).bit_length()+1 # Chai Wah Wu, Jul 07 2022

Formula

Equals A053398(2, n).

G.f.: (1+x)/x^2 * Sum(k>=1, x^(2^k)/(1-x^(2^k))). - Ralf Stephan, Apr 12 2002

a(n) = A136480(n+1). - Reinhard Zumkeller, Dec 31 2007

a(n) = A007814(n + 2 - n mod 2). - James Spahlinger, Oct 11 2013, corrected by Charles R Greathouse IV, Oct 14 2013

a(2n) = a(2n+1). 1 <= a(n) <= log_2(n+2). - Charles R Greathouse IV, Oct 14 2013

a(n) = A007814(n+1)+A007814(n+2).

a(n) = (-1)^n * A094267(n). - Michael Somos, May 11 2014

a(n) = A007814(floor(n/2)+1). - Chai Wah Wu, Jul 07 2022

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

Extensions

Definition simplified by N. J. A. Sloane, Aug 27 2016