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User: Giuseppe Ciacco

Giuseppe Ciacco has authored 2 sequences.

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

Giuseppe Ciacco, Feb 16 2024

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).

			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.

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.
Wikipedia, Collatz Conjecture.
OEIS Wiki, 3x+1 problem.

Cf. A006666, A368136.

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 *)

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

A368136 Numbers k for which a generalized Collatz trajectory (x / k if k divides x, x + ceiling(x / k) otherwise) has non-elementary loops starting from a positive integer x_0 < k^2.

Original entry on oeis.org

3, 4, 6, 9, 10, 15, 16, 17, 20, 23, 24, 27, 29, 31, 48, 54, 57, 78, 85, 94, 111, 118, 123, 127, 129, 134, 136, 171, 172, 225, 368, 419, 540, 547, 706, 744, 1112, 1148, 1169, 1229, 1308, 1403, 1545, 1782, 1869, 1926, 1939

Offset: 1

Giuseppe Ciacco, Dec 13 2023

For a given k, define the generalized Collatz trajectory starting at x_0 > 0 as follows:
  x_(i+1) = x_(i) / k if k divides x_(i);
  x_(i+1) = x_(i) + ceiling(x_(i) / k) otherwise.
For k = 2, this is equivalent to the Collatz step x -> x/2 or (3x + 1)/2.
We call a loop an 'elementary loop' if it contains 1 as a term and otherwise a 'non-elementary loop'. The loop containing 1 consists of the terms 1, 4, 2, 1 for k = 2, or 1, 2, ..., k, 1 for other k.
k^2 has been chosen as an arbitrary boundary, giving more terms of the (limiting) sequence (i.e., the unknown sequence that would result if no boundary were used) than using 2*k, 3*k, or similar boundaries. It is unknown whether there are values of k for which non-elementary loops exist only for values greater than k^2.
It is also unknown whether there are values of k and x_0 for which trajectories do not contain any loop. Such values would be terms of the sequence only if there are also non-elementary loops.

			k = 3 is a term since it has a non-elementary loop starting from x_0 = 7:
  7, 10, 14, 19, 26, 35, 47, 63, 21, 7, ...
k = 2 is not a term since it has no non-elementary loops starting from x_0 < 4.

Walter Carnielli, Some natural generalizations of the Collatz Problem, Applied Mathematics E-Notes 15 (2015): 207-215.
Wikipedia, Collatz Conjecture.
Wikipedia, Floyd's cycle detection algorithm.
OEIS Wiki, 3x+1 problem.

Cf. A006370.

See A033478 for an example of a trajectory (based on the 3x + 1 formulation) with k = 2 and x_0 = 3, ending in an elementary loop.

def containsloops(k):
    for x_ in range(k, k*k):
        s = 0
        x = x_
        m = x
        while x != 1 and s <= m:
            d, r = divmod(x, k)
            x = d if r == 0 else d + x + 1
            s += 1
            m = max(m, x)
        if s > m and x > k:
            return True
    return False
print([k for k in range(1, 100) if containsloops(k)])

a(43)-a(45) from Giuseppe Ciacco, Feb 05 2024

a(46)-a(48) from Giuseppe Ciacco, Feb 14 2024

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User: Giuseppe Ciacco

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.

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