A370350 Number of steps to go from n to 1 in a variant of the Collatz iteration x -> (x / 5 if 5 divides x, x + (x+(5-(x mod 5)))/5 otherwise), or -1 if 1 is never reached.

0, 4, 3, 2, 1, 7, 17, 6, 16, 5, 15, 5, 5, 14, 4, 4, 13, 11, 11, 3, 12, 10, 10, 20, 2, 11, 9, 9, 19, 8, 69, 10, 8, 8, 18, 17, 18, 68, 9, 7, 7, 8, 77, 16, 17, 67, 8, 17, 8, 6, 7, 76, 15, 7, 16, 66, 7, 16, 7, 6, 75, 6, 75, 14, 6, 6, 16, 65, 6, 15, 6, 15, 24, 74

Offset

1,2

Comments

Because ceiling(a/b) = (a+(b-(a mod b)))/b, this sequence equals the total number of steps to reach 1 in a variant of the Collatz problem using the iteration f(x) := x/k if k divides x, x+ceiling(x/k) otherwise.

Specifically, here we set k=5 because it is the next integer (after 2) that is not in A368136; i.e. the iteration used here has no loops for starting values up to 5*5=25, apart from the loop containing 1.

It is not known if there exists n such that a(n) = -1 (either by iteration reaching a non-elementary loop which implies n>5*5, or by iteration growing without bound).

Links

Giuseppe Ciacco, Table of n, a(n) for n = 1..10000

Walter Carnielli, Some natural generalizations of the Collatz Problem, Applied Mathematics E-Notes 15 (2015): 207-215.

OEIS Wiki, 3x+1 problem.

Wikipedia, Collatz Conjecture.

Example

For n = 11, the following trajectory is obtained:
  11, 14, 17, 21, 26, 32, 39, 47, 57, 69, 83, 100, 20, 4, 5, 1
which requires 15 steps to reach 1, therefore a(11) = 15.

Mathematica

s={}; Do[c=0; a=n; While[a>1, If[Divisible[a, 5], a=a/5, a=a+Ceiling[a/5]]; c++]; AppendTo[s, c], {n, 74}]; s (* James C. McMahon, Feb 28 2024 *)

Prog

(Python)
def a(n, C = 5):
    s = 0
    while n > 1:
        d, r = divmod(n, C)
        n = n + 1 + d if r else d
        s += 1
    return s
print([a(n) for n in range(1, 75)])
# Giuseppe Ciacco and Robert Munafo, Mar 25 2024

Crossrefs

Cf. A006666, A368136.

Sequence in context: A194738 A194750 A388583 * A194743 A113778 A295673

Adjacent sequences: A370347 A370348 A370349 * A370351 A370352 A370353

Keyword

nonn,easy,look

Author

Giuseppe Ciacco, Feb 16 2024

Status

approved