login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A160198 a(n) = min(A122458(n), A159885(n)). 9
2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 4, 1, 3, 1, 2, 1, 2, 1, 3, 1, 3, 1, 3, 1, 2, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 4, 1, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Let f(2n+1) = A000265(3n+2) be defined as in A159885. Then a(n) is the least number k of iterations such that either f^k(2n+1) < 2n+1 or A000120(f^k(2n+1)) < A000120(2n+1).

Using induction, one can prove that the Collatz (3x+1)-conjecture follows from the finiteness of a(n) for every n. - Vladimir Shevelev, May 05 2009

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences related to 3x+1 (or Collatz) problem

MAPLE

A000265 := proc(n) option remember ; local a; a := n ; while a mod 2 = 0 do a := a/2 ; end do; a; end proc:

f := proc(n) local m ; m := (n-1)/2 ; A000265(3*m+2) ; end:

A000120 := proc(n) local d; add(d, d=convert(n, base, 2)) ; end proc:

A159885 := proc(n) local k, twon1; k := 0 ; twon1 := 2*n+1 ; while ( A000120(twon1) > A000120(n) ) do twon1 := f(twon1) ; k := k+1 ; end do; k ; end proc:

A122458 := proc(n) local tx1, a; a := 0 ; tx1 := 2*n+1 ; while tx1 >= 2*n+1 do if tx1 mod 2 = 0 then tx1 := tx1/2 ; else tx1 := 3*tx1+1 ; a := a+1 ; fi; end do; a ; end proc:

A160198 := proc(n) min(A159885(n), A122458(n)) ; end: seq(A160198(n), n=1..130) ; # R. J. Mathar, May 15 2009

PROG

(PARI)

f(n) = ((3*((n-1)/2))+2)/A006519((3*((n-1)/2))+2);  \\ Defined for odd n only. Cf. A075677.

A006519(n) = (1<<valuation(n, 2));

A160198(n) = { my(u = (n+n+1), w = hammingweight(u), k=0); while((u >= (n+n+1))&&(hammingweight(u) >= w), k++; u = f(u)); (k); }; \\ Antti Karttunen, Sep 22 2018

CROSSREFS

Cf. A000120, A075677, A122458, A159885, A159945, A160267.

Sequence in context: A266871 A060500 A187284 * A207709 A131718 A131017

Adjacent sequences:  A160195 A160196 A160197 * A160199 A160200 A160201

KEYWORD

nonn,look

AUTHOR

Vladimir Shevelev, May 04 2009

EXTENSIONS

a(1) corrected and sequence extended by R. J. Mathar, May 15 2009

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 9 00:32 EST 2019. Contains 329871 sequences. (Running on oeis4.)