A085058 a(n) = A001511(n+1) + 1.

2, 3, 2, 4, 2, 3, 2, 5, 2, 3, 2, 4, 2, 3, 2, 6, 2, 3, 2, 4, 2, 3, 2, 5, 2, 3, 2, 4, 2, 3, 2, 7, 2, 3, 2, 4, 2, 3, 2, 5, 2, 3, 2, 4, 2, 3, 2, 6, 2, 3, 2, 4, 2, 3, 2, 5, 2, 3, 2, 4, 2, 3, 2, 8, 2, 3, 2, 4, 2, 3, 2, 5, 2, 3, 2, 4, 2, 3, 2, 6, 2, 3, 2, 4, 2, 3, 2, 5, 2, 3, 2, 4, 2, 3, 2, 7, 2, 3, 2, 4, 2, 3, 2, 5, 2

Offset: 0

Author

N. J. A. Sloane, Aug 11 2003

Keywords

• nonn
• easy

Comments

Number of divisors of 2n+2 of the form 2^k. - Giovanni Teofilatto, Jul 25 2007

Number of steps for iteration of map x -> (3/2)*ceiling(x) to reach an integer when started at 2*n+1.

Also number of steps for iteration of map x -> (3/2)*floor(x) to reach an integer when started at 2*n+3. - Benoit Cloitre, Sep 27 2003

The first time that a(n) = e+1 is when n is of the form 2^e - 1. - Robert G. Wilson v, Sep 28 2003

Let 2^k(n) = largest power of 2 dividing tangent number A000182(n). Then a(n-1) = 2*n - k(n). - Yasutoshi Kohmoto, Dec 23 2006

a(n) is the number of integers generated by b(i+1) = (3+2n)*(b(i) + b(i-1))/2, following these two initial values, b(0) = b(1) = 1. Thereafter only non-integers are generated. - Richard R. Forberg, Nov 09 2014

a(n) is the 2-adic valuation of 4*n+4, which is equal to the number of trailing 1-bits of 4*n+3 in binary. - Ruud H.G. van Tol, Sep 11 2023

Links

• Antti Karttunen, Table of n, a(n) for n = 0..16383
• J. C. Lagarias and N. J. A. Sloane, Approximate squaring, Experimental Math., 13 (2004), 113-128.

Crossrefs

Cf. A001511, A085060, A007814, A105723, A000182.

Programs

Magma

[Valuation(n+1, 2)+2: n in [0..100]]; // Vincenzo Librandi, Jan 16 2016

Maple

f := x->(3/2)*ceil(x); g := proc(n) local t1,c; global f; t1 := f(n); c := 1; while not type(t1, 'integer') do c := c+1; t1 := f(t1); od; RETURN([c,t1]); end;
a := n ->  A001511(n+1) + 1: A001511 := n -> padic[ordp](2*n, 2): seq(a(n), n=0..104); # Johannes W. Meijer, Dec 22 2012

Mathematica

g = 3 Ceiling[ # ]/2 &; f[n_?OddQ] := Length @ NestWhileList[ g, g[n], !IntegerQ[ # ] & ]; Table[ f[n], {n, 1, 210, 2}]

PARI

A085058(n)=if(n<0,0,c=2*n+7/2; x=0; while(frac(c)>0,c=3/2*floor(c); x++); x) \\ Benoit Cloitre, Sep 27 2003

PARI

A085058(n)=if(n<0,0,c=(2*n+1)*3/2; x=1; while(frac(c)>0,c=3/2*ceil(c); x++); x) \\ Benoit Cloitre, Sep 27 2003

PARI

a(n) = valuation(n+1,2)+2; \\ Michel Marcus, Jan 15 2016

Python

def A085058(n): return (~(n+1) & n).bit_length()+2 # Chai Wah Wu, Apr 14 2023

Formula

a(n) = A007814(3^(n+1) - (-1)^(n+1)) = A007814(A105723(n+1)). - Reinhard Zumkeller, Apr 18 2005

a(n) = A001511(n+1) + 1 = A001511(2*n+2). - Ray Chandler, Jul 29 2007

a(n) = A007814(5^(n+1) - 1). - Ivan Neretin, Jan 15 2016

a(n) = A007814(4*(n+1)) = A007814(n+1) + 2. - Ruud H.G. van Tol, Sep 11 2023

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3. - Amiram Eldar, Sep 13 2024

Extensions

Edited by Franklin T. Adams-Watters, Dec 09 2013