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A160322 a(n) = min(A160198(n), A160267(n)). 2
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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).

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

LINKS

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

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

FORMULA

a(n) = min(A122458(n), A159885(n), A160266(n)).  - Antti Karttunen, Sep 25 2018

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));

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

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

CROSSREFS

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

Sequence in context: A231425 A136713 A224782 * A198293 A043281 A320267

Adjacent sequences:  A160319 A160320 A160321 * A160323 A160324 A160325

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, May 08 2009, May 11 2009

EXTENSIONS

a(1) corrected and sequence extended by Antti Karttunen, Sep 25 2018

STATUS

approved

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Last modified November 16 22:05 EST 2019. Contains 329208 sequences. (Running on oeis4.)