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A290101 a(n) = dropping time for the modified Collatz problem, where x -> 3x+1 if x is odd, and x -> either x/2 or 3x+1 if x is even (minimal number of any such steps to reach a lower number than the starting value n); a(1) = 0 by convention. 3
0, 1, 6, 1, 3, 1, 11, 1, 3, 1, 8, 1, 3, 1, 11, 1, 3, 1, 6, 1, 3, 1, 8, 1, 3, 1, 16, 1, 3, 1, 19, 1, 3, 1, 6, 1, 3, 1, 11, 1, 3, 1, 8, 1, 3, 1, 16, 1, 3, 1, 6, 1, 3, 1, 8, 1, 3, 1, 11, 1, 3, 1, 19, 1, 3, 1, 6, 1, 3, 1, 13, 1, 3, 1, 8, 1, 3, 1, 13, 1, 3, 1, 6, 1, 3, 1, 8, 1, 3, 1, 11, 1, 3, 1, 13, 1, 3, 1, 6, 1, 3, 1, 16, 1, 3, 1, 8, 1, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

In contrast to the "3x+1" problem (see A006577, A102419), here you are free to choose either step if x is even. The sequence counts the minimum number of optimally chosen steps which leads to a value smaller than the value we started from.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

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

FORMULA

a(n) <= A102419(n).

a(n) <= A127885(n) [apart from any hypothetical -1's in A127885].

EXAMPLE

Starting from n = 27, the following is a shortest path leading to a value smaller than 27: 27 -> 82 -> 41 -> 124 -> 373 -> 1120 -> 560 -> 280 -> 140 -> 70 -> 35 -> 106 -> 53 -> 160 -> 80 -> 40 -> 20. It has 16 steps, thus a(27) = 16. Note the 3x+1 step from 124 to 373 which is not allowed in the ordinary Collatz problem.

PROG

(PARI) A290101(n) = { if(1==n, return(0)); my(S, k); S=[n]; k=0; while( S[1]>=n, k++; S=vecsort( concat(apply(x->3*x+1, S), apply(x->x\2, select(x->x%2==0, S) )), , 8);  ); k } \\ After Max Alekseyev's code for A127885

(Python)

from sympy import flatten

def ok(n, L):

    for i in L: return 1 if i<n else 0

def a(n):

    if n==1: return 0

    L=[n]

    i=0

    while not ok(n, L):

        L=sorted(list(set(flatten([[3*k + 1, k/2] if k%2==0 else 3*k + 1 for k in L]))))

        i+=1

    return i

print (map(a, range(1, 121))) # Indranil Ghosh, Aug 31 2017

CROSSREFS

Cf. A127885, A290100, A290102, A000012 (even bisection).

Differs from A102419 for the first time at n=27, where a(27) = 16, while A102419(27) = 96.

Sequence in context: A154911 A152935 A253686 * A102419 A074193 A074453

Adjacent sequences:  A290098 A290099 A290100 * A290102 A290103 A290104

KEYWORD

nonn

AUTHOR

Antti Karttunen, Aug 20 2017

STATUS

approved

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Last modified October 18 14:44 EDT 2019. Contains 328161 sequences. (Running on oeis4.)