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

Author

Nicholas Drozd, Dec 03 2021

Keywords

• nonn
• easy
• look

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.

Examples

			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.
		

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.

Crossrefs

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

Cf. also A006370.

Programs

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 *)

PARI

a(n) = my(d=divrem(n, 3)); if (d[2], a(5*d[1]+d[2]+3), n); \\ Michel Marcus, Dec 05 2021

Python

def a(n):
    while True:
        quot, rem = divmod(n, 3)
        if rem == 0:
            return n
        n = (5 * quot) + rem + 3

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

Extensions

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