A330732 a(n) is the smallest number whose odd hailstone sequence has length n (including 1 and itself) and is in descending order.

1, 5, 13, 17, 45, 241, 321, 1713, 9137, 24365, 36033, 173261, 231681, 630033, 1642769, 4380717, 11715629, 31241677, 41655569, 111081517, 148108689, 789913009, 1053217345, 1404289793, 3744772781, 9986060749, 13314747665, 35505993773, 94682650061, 128599099649

Offset

1,2

Comments

The hailstone (Collatz) sequence of a number is defined by repeated operations of f(x), where f(x)=x/2 if x is even, and f(x)=3x+1 if x is odd. The odd numbers of a number's hailstone sequence are also called its odd hailstone sequence.

The odd hailstone sequence of 45 [45, 17, 13, 5, 1] contains the first five terms of this sequence.

Note that a(n+1) isn't necessarily greater than a(n).

Links

Kevin P. Thompson, Table of n, a(n) for n = 1..36

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

Example

Consider a(8): [1713, 1285, 241, 181, 17, 13, 5, 1]; the odd hailstone sequence of 1713 is descending and has length 8. 1713 is the lowest such number.

Mathematica

With[{s = Array[If[AllTrue[Differences@ #, # < 0 &], Length@ #, 0] &@ Select[NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, #, # > 1 &], OddQ] &, 10^5]}, Array[FirstPosition[s, #][[1]] &, Max@ s]] (* Michael De Vlieger, Jan 01 2020 *)

Prog

(PARI) is(k, n) = {my(x=k, v=List([])); while(x>1, if(x%2==0, x=x/2, listput(v, x); x=3*x+1)); 1+#v==n&&v==vecsort(v, , 4); }
a(n) = {k=1; while(is(k, n)==0, k+=2); k; } \\ Jinyuan Wang, Dec 31 2019

Crossrefs

Cf. A078719, A092893, A182078.

Sequence in context: A120130 A087484 A019382 * A276260 A350379 A145040

Adjacent sequences: A330729 A330730 A330731 * A330733 A330734 A330735

Keyword

nonn

Author

Xiangyu Chen, Dec 28 2019

Extensions

a(23)-a(30) from Kevin P. Thompson, Jul 23 2022

Status

approved