A349877 -id:A349877 - cp's OEIS frontend

A353314 If n is of the form 3k, then a(n) = n, and if n is of the form 3k+r, with r = 1 or 2, then a(n) = 5*k + 3 + r.

0, 4, 5, 3, 9, 10, 6, 14, 15, 9, 19, 20, 12, 24, 25, 15, 29, 30, 18, 34, 35, 21, 39, 40, 24, 44, 45, 27, 49, 50, 30, 54, 55, 33, 59, 60, 36, 64, 65, 39, 69, 70, 42, 74, 75, 45, 79, 80, 48, 84, 85, 51, 89, 90, 54, 94, 95, 57, 99, 100, 60, 104, 105, 63, 109, 110, 66, 114, 115, 69, 119, 120, 72, 124, 125, 75, 129, 130

Offset: 0

Antti Karttunen, Apr 14 2022

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Nicholas Drozd, A Busy Beaver Champion Derived from Scratch
Nicholas Drozd, Feedback to Doron Zeilberger's opinion #155, Jan. 4, 2022.
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A353313 (variant), A349876 (the first multiple of 3 reached when iterating this sequence), A349877 (number of iterations to reach the first multiple of 3), A353327 (A102899).

Array[If[#2 == 0, #1, 5 #1 + 3 + #2 & @@ QuotientRemainder[#1, 3]] & @@ {#, Mod[#, 3]} &, 78, 0] (* Michael De Vlieger, Apr 14 2022 *)

A353314(n) = { my(r=(n%3)); if(!r,n,((5*((n-r)/3)) + r + 3)); };

a(n) = n + A353327(n) = n + A102899(3+n).
From Chai Wah Wu, Jul 27 2022: (Start)
a(n) = 2*a(n-3) - a(n-6) for n > 5.
G.f.: x*(x^3 + 3*x^2 + 5*x + 4)/(x^6 - 2*x^3 + 1). (End)

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.

Original entry on oeis.org

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

Nicholas Drozd, Dec 03 2021

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.

			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.

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.

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

Cf. also A006370.

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 *)

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

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

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

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

A353311 Number of distinct terms encountered when A353313 is iterated, or -1 if the iteration never reaches a finite cycle.

Original entry on oeis.org

1, 4, 104, 4, 4, 103, 105, 5, 104, 4, 103, 4, 5, 106, 4, 103, 6, 105, 106, 103, 3, 6, 108, 3, 105, 3, 104, 5, 3, 6, 104, 108, 110, 5, 103, 3, 6, 14, 116, 107, 3, 54, 5, 57, 3, 103, 7, 104, 6, 3, 6, 106, 9, 106, 107, 109, 12, 104, 7, 103, 3, 115, 5, 7, 13, 115, 109, 118, 21, 3, 53, 5, 106, 121, 56, 3, 59, 124, 105

Offset: 0

Antti Karttunen, Apr 13 2022

			Starting from n=2 and iterating A353313, we obtain the following 104 terms [2, 5, 10, 19, 34, 59, 100, 169, 284, 475, 794, 1325, 2210, 3685, 6144, 2048, 3415, 5694, 1898, 3165, 1055, 1760, 2935, 4894, 8159, 13600, 22669, 37784, 62975, 104960, 174935, 291560, 485935, 809894, 1349825, 2249710, 3749519, 6249200, 10415335, 17358894, 5786298, 1928766, 642922, 1071539, 1785900, 595300, 992169, 330723, 110241, 36747, 12249, 4083, 1361, 2270, 3785, 6310, 10519, 17534, 29225, 48710, 81185, 135310, 225519, 75173, 125290, 208819, 348034, 580059, 193353, 64451, 107420, 179035, 298394, 497325, 165775, 276294, 92098, 153499, 255834, 85278, 28426, 47379, 15793, 26324, 43875, 14625, 4875, 1625, 2710, 4519, 7534, 12559, 20934, 6978, 2326, 3879, 1293, 431, 720, 240, 80, 135, 45, 15] before the iteration returns to 5 again, therefore a(2) = 104.

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

Cf. A349877, A353310, A353312, A353313.

A353313(n) = { my(r=(n%3)); if(!r,n/3,((5*((n-r)/3)) + r + 3)); };
A353311(n) = { my(visited = Map()); for(j=0, oo, if(mapisdefined(visited, n), return(j), mapput(visited, n, j)); n = A353313(n)); };

a(n) = A353310(n) + A353312(n).

a(n) > A349877(n) for all n.

A353312 Size of the finite cycle eventually reached by iterating A353313, or -1 if no finite cycle is ever reached.

Original entry on oeis.org

1, 4, 103, 4, 4, 103, 103, 3, 103, 4, 103, 3, 4, 103, 3, 103, 6, 103, 103, 103, 3, 3, 103, 3, 103, 3, 103, 4, 3, 6, 103, 103, 103, 3, 103, 3, 4, 3, 103, 103, 3, 3, 3, 3, 3, 103, 3, 103, 6, 3, 6, 103, 6, 103, 103, 103, 6, 103, 3, 103, 3, 103, 3, 3, 3, 103, 103, 103, 6, 3, 3, 3, 103, 103, 3, 3, 3, 103, 103, 3, 103

Offset: 0

Antti Karttunen, Apr 13 2022

nonn

			Starting from n=2 and iterating A353313, we obtain the following 104 terms [2, 5, 10, 19, 34, 59, 100, 169, 284, 475, 794, 1325, 2210, 3685, 6144, 2048, 3415, 5694, 1898, 3165, 1055, 1760, 2935, 4894, 8159, 13600, 22669, 37784, 62975, 104960, 174935, 291560, 485935, 809894, 1349825, 2249710, 3749519, 6249200, 10415335, 17358894, 5786298, 1928766, 642922, 1071539, 1785900, 595300, 992169, 330723, 110241, 36747, 12249, 4083, 1361, 2270, 3785, 6310, 10519, 17534, 29225, 48710, 81185, 135310, 225519, 75173, 125290, 208819, 348034, 580059, 193353, 64451, 107420, 179035, 298394, 497325, 165775, 276294, 92098, 153499, 255834, 85278, 28426, 47379, 15793, 26324, 43875, 14625, 4875, 1625, 2710, 4519, 7534, 12559, 20934, 6978, 2326, 3879, 1293, 431, 720, 240, 80, 135, 45, 15] before the iteration returns to 5 again, in other words, forming a finite cycle of length 103, therefore a(2) = 103.

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

Cf. A349877, A353310, A353311, A353313.

A353313(n) = { my(r=(n%3)); if(!r,n/3,((5*((n-r)/3)) + r + 3)); };
A353312(n) = { my(visited = Map()); for(j=1, oo, if(mapisdefined(visited, n), return(j-mapget(visited, n)), mapput(visited, n, j)); n = A353313(n)); };

a(n) = A353311(n) - A353310(n).

cp's OEIS Frontend

A353314 If n is of the form 3k, then a(n) = n, and if n is of the form 3k+r, with r = 1 or 2, then a(n) = 5*k + 3 + r.

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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.

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A353311 Number of distinct terms encountered when A353313 is iterated, or -1 if the iteration never reaches a finite cycle.

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A353312 Size of the finite cycle eventually reached by iterating A353313, or -1 if no finite cycle is ever reached.

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