

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
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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



