A349876 If n is divisible by 3, a(n) = n; otherwise n = 3k + r with r in {1, 2} and a(n) = a(5k + r + 3). a(n) = -1 if no multiple of three will be ever reached by iterating A353314.

0, 9, 6144, 3, 9, 6144, 6, 75, 15, 9, 6144, 60, 12, 24, 75, 15, 144, 30, 18, 6144, 60, 21, 39, 69, 24, 75, 45, 27, 84, 144, 30, 54, 159, 33, 6144, 60, 36, 309, 519, 39, 69, 1560, 42, 210, 75, 45, 225, 135, 48, 84, 144, 51, 150, 90, 54, 159, 450, 57, 99, 6144, 60, 294, 105, 63

Offset

0,2

Comments

It is not known if this function is total, that is, if a(n) is well-defined for all n (this is the reason for the escape clause in the definition).

This is a so-called "Collatz-like" function, because A353313 and A353314 have some similarity to the Collatz function A006370.

The sequence can be implemented with a Turing machine with 4 states and 2 colors.

Links

Antti Karttunen, Table of n, a(n) for n = 0..19683

Nick Drozd, A Busy Beaver Champion Derived from Scratch.

Shawn Ligocki, Collatz-like behavior of Busy Beavers.

Pascal Michel, Busy Beaver competition and Collatz-like problems, Arch. Math. Logic (1993), 32:351-367.

Formula

If A010872(n) = 0 [when n is a multiple of 3], a(n) = n, otherwise a(n) = a(A353314(n)). [Here one may also use A353313 instead of A353314] - Antti Karttunen, Apr 14 2022

Example

a(1) = a(3*0 + 1) = a(5*0 + 1 + 3) = a(4)
a(4) = a(3*1 + 1) = a(5*1 + 1 + 3) = a(9)
a(9) = 9
Trajectory of a(1): 4, 9
a(7) = a(3*2 + 1) = a(5*2 + 1 + 3) = a(14)
a(14) = a(3*4 + 2) = a(5*4 + 2 + 3) = a(25)
a(25) = a(3*8 + 1) = a(5*8 + 1 + 3) = a(44)
a(44) = a(3*14 + 2) = a(5*14 + 2 + 3) = a(75)
a(75) = 75
Trajectory of a(7): 14, 25, 44, 75
Trajectory of a(2): 5, 10, 19, 34, 59, 100, 169, 284, 475, 794, 1325, 2210, 3685, 6144.

Mathematica

a[n_] := a[n] = Module[{qr = QuotientRemainder[n, 3]}, If[qr[[2]] == 0, n, a[5*qr[[1]] + qr[[2]] + 3]]]; Array[a, 64, 0] (* Amiram Eldar, Jan 04 2022 *)

Prog

(Python)
def a(n):
    while True:
        quot, rem = divmod(n, 3)
        if rem == 0:
            return n
        n = (5 * quot) + rem + 3
(PARI) a(n) = my(d=divrem(n, 3)); if (d[2], a(5*d[1]+d[2]+3), n); \\ Michel Marcus, Dec 05 2021

Crossrefs

Cf. A010872, A349877, A349896 (record values), A353313, A353314.

Cf. also A006370.

Sequence in context: A185821 A133248 A125542 * A349896 A013744 A060761

Adjacent sequences: A349873 A349874 A349875 * A349877 A349878 A349879

Keyword

nonn,easy,look

Author

Nicholas Drozd, Dec 03 2021

Extensions

Formal escape clause added to the definition by Antti Karttunen, Apr 14 2022

Status

approved