This site is supported by donations to The OEIS Foundation.

 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!)
 A160322 a(n) = min(A160198(n), A160267(n)). 2

%I

%S 2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,

%T 1,1,1,1,2,1,2,1,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,

%U 3,1,2,1,2,1,1,1,1,1,1,1,2,1,9,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2

%N a(n) = min(A160198(n), A160267(n)).

%C Let f be defined as in A159885. Then a(n) is the least k such that either f^k(2n+1))<2n+1 or A000120(f^k(2n+1)) < A000120(2n+1) or A006694((f^k(2n+1)-1)/2) < A006694(n).

%C In connection with A160198, A160267, A160322 we pose a new (3x+1)-problem: does there exist a finite number of sequences A_i(n), i=1,...,T, such that: 1) A_i(0)=0 and A_i(n)>0 for n>=1; 2) if B_i(n) denotes the least k for which A_i(n)>A_i((f^k(2n+1)-1)/2), then B(n)=min_{i=1,...,T}B_i(n)=1 for every n>=1? Note that this problem is weaker than (3x+1)-Collatz problem. Indeed, if the Collatz conjecture is true, then there exist nonnegative sequences A(n) for which A(0)=0 and A(n)>A((f(2n+1)-1)/2) for every n>=1 (see A160348). - _Vladimir Shevelev_, May 15 2009

%H Antti Karttunen, <a href="/A160322/b160322.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>

%F a(n) = min(A122458(n), A159885(n), A160266(n)). - _Antti Karttunen_, Sep 25 2018

%o (PARI)

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

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

%o A006694(n) = (sumdiv(2*n+1, d, eulerphi(d)/znorder(Mod(2, d))) - 1); \\ From A006694

%o A160322(n) = { my(v=A006694(n), u = (n+n+1), w = hammingweight(u), k=0); while((u >= (n+n+1))&&(hammingweight(u) >= w)&&(A006694((u-1)/2) >= v), k++; u = f(u)); (k); }; \\ _Antti Karttunen_, Sep 25 2018

%Y Cf. A000120, A006694, A122458, A159885, A159945, A160198, A160266, A160267.

%K nonn

%O 1,1

%A _Vladimir Shevelev_, May 08 2009, May 11 2009

%E a(1) corrected and sequence extended by _Antti Karttunen_, Sep 25 2018

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.

Last modified December 16 03:14 EST 2019. Contains 330013 sequences. (Running on oeis4.)