A075680 - OEIS

A075680

For odd numbers 2n-1, the minimum number of iterations of the reduced Collatz function R required to yield 1. The function R is defined as R(k) = (3k+1)/2^r, with r as large as possible.

0, 2, 1, 5, 6, 4, 2, 5, 3, 6, 1, 4, 7, 41, 5, 39, 8, 3, 6, 11, 40, 9, 4, 38, 7, 7, 2, 41, 10, 10, 5, 39, 8, 8, 3, 37, 42, 3, 6, 11, 6, 40, 1, 9, 9, 33, 4, 38, 43, 7, 7, 31, 12, 36, 41, 24, 2, 10, 5, 10, 34, 15, 39, 15, 44, 8, 8, 13, 32, 13, 3, 37, 42, 42, 6, 3, 11, 30, 11, 18, 35, 6, 40, 23

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OFFSET

1,2

COMMENTS

See A075677 for the function R applied to the odd numbers once. The 3x+1 conjecture asserts that a(n) is a finite number for all n. The function R applied to the odd numbers shows the essential behavior of the 3x+1 iterations.

Bisection of A006667. - T. D. Noe, Jun 01 2006

LINKS

T. D. Noe, Table of n, a(n) for n = 1..5000

EXAMPLE

a(4) = 5 because 7 is the fourth odd number and 5 iterations are needed: R(R(R(R(R(7)))))=1.

MATHEMATICA

nextOddK[n_] := Module[{m=3n+1}, While[EvenQ[m], m=m/2]; m]; (* assumes odd n *) Table[m=n; cnt=0; If[n>1, While[m=nextOddK[m]; cnt++; m!=1]]; cnt, {n, 1, 200, 2}]

PROG

(Haskell)

a075680 n = snd $ until ((== 1) . fst)

(\(x, i) -> (a000265 (3 * x + 1), i + 1)) (2 * n - 1, 0)

-- Reinhard Zumkeller, Jan 08 2014

(Perl)

sub a {

my $v = 2 * shift() - 1;

my $c = 0;

until (1 == $v) {

$v = 3 * $v + 1;

$v /= 2 until ($v & 1);