A087230 a(n) is the 2-adic valuation of 6*n + 4.

2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 8, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1

Offset: 0

Author

Labos Elemer, Aug 28 2003

Keywords

• nonn
• easy

Comments

In Collatz-algorithm if initiated with m=odd value, the first 3x+1 step is followed by a(n) step corresponding to division by 2. Compare to A085058 and A087229. Each 2nd term is either =1 or equals corresponding term of A087229, depending on whether the odd number congruent to 1 or 3 modulo 4.

From K. G. Stier, Aug 19 2014: (Start)

Sequence exhibits a "pseudo" ruler function (A001511) behavior. It is similar to the latter in repeating equal terms m>0 after each 2^m steps. However, the first occurrence of m in the mentioned ruler function is simply at n=log_2(m), while in the given sequence this property develops two distinct (odd and even) strands:

First occurrence of

m=1 at a(1); m=2 at a(0)

m=3 at a(6); m=4 at a(2)

m=5 at a(26); m=6 at a(10)

m=7 at a(106); m=8 at a(42)

m=9 at a(426); m=10 at a(170)

...

where values n in the odd strand (1,6,26,106,426,...) equal sequence A020989, and values n in the even strand (0,2,10,42,170,...) equal sequence A020988. (End)

Examples

			n=85: m = 6*85+4 = 514 and Collatz-iteration goes on by one dividing step, a(85)=1.
		

Links

• Kenny Lau, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A007814, A016957, A085058, A087229, A020988, A020989, A061547.

Programs

Maple

a:= n-> padic[ordp](6*n+4, 2):
seq(a(n), n=0..120);  # Alois P. Heinz, Mar 16 2021

Mathematica

Table[Part[Part[FactorInteger[6*w+4], 1], 2], {w, 0, 100}]
Table[IntegerExponent[6*n + 4, 2], {n, 0, 100}] (* Amiram Eldar, Jan 27 2022 *)

PARI

forstep(n=0, 1000, 1, m=6*n+4; print1(valuation(m, 2), ", ") ) \\ K. G. Stier, Aug 19 2014

Perl

sub a {
  my $nv= ((shift() << 1) | 1);
  my $bp= 1;
  while (($nv & 1) xor ($nv & 2)) {
    $nv>>= 1;
    $bp++;
  }
  return $bp;
} # Ruud H.G. van Tol, Nov 16 2021

Python

n=100; N=3*n+2; val=[1]*(N+1); exp=2
while exp <= N:
    for j in range(exp,N+1,exp): val[j] += 1
    exp *= 2
for i in range(n+1): print(i,val[3*i+2])
# Kenny Lau, Jun 09 2018

Python

def A087230(n): return (~(m:=6*n+4) & m-1).bit_length() # Chai Wah Wu, Jul 02 2022

Formula

a(n) = A007814(A016957(n)). - Michel Marcus, Jan 27 2022

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Sep 10 2024

Extensions

a(0) = 2 prepended by Andrey Zabolotskiy, Jan 27 2022, based on Ihar Senkevich's contribution