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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

MATHEMATICA

a[n_] := Module[{u=2n+1, w, k=0}, w = DigitCount[u, 2, 1]; While[u >= 2n+1 && DigitCount[u, 2, 1] >= w, k++; u = (3(u-1)/2+2)/2^IntegerExponent[ (3(u-1)/2+2), 2]]; k];

Array[a, 105] (* Jean-Fran├žois Alcover, Apr 16 2020, after Antti Karttunen *)

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: A331290 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

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Last modified September 18 01:39 EDT 2021. Contains 347504 sequences. (Running on oeis4.)