A353313 -id:A353313 - cp's OEIS frontend

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

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.

A353310 Number of terms encountered when iterating A353313, before reaching the first term that is a part of a finite cycle, or -1 if no finite cycle is ever reached.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 2, 2, 1, 0, 0, 1, 1, 3, 1, 0, 0, 2, 3, 0, 0, 3, 5, 0, 2, 0, 1, 1, 0, 0, 1, 5, 7, 2, 0, 0, 2, 11, 13, 4, 0, 51, 2, 54, 0, 0, 4, 1, 0, 0, 0, 3, 3, 3, 4, 6, 6, 1, 4, 0, 0, 12, 2, 4, 10, 12, 6, 15, 15, 0, 50, 2, 3, 18, 53, 0, 56, 21, 2, 3, 0, 2, 5, 19, 0, 0, 62, 1, 59, 2, 2, 27, 65, 6, 5, 5, 8, 90, 8

Offset: 0

Antti Karttunen, Apr 14 2022

nonn

			For n = 1, when iterating with A353313, we obtain 1 -> 4 -> 9 -> 3 -> 1 -> etc, thus 1 itself is included in the closed cycle, and therefore a(1) = 0.
For n = 6, when iterating with A353313, we obtain 6 -> 2 -> 5 -> ..., and after 103 more iterations we obtain 5 again (see examples in A353311 and A353312), thus only the two initial numbers, 6 and 2 are outside of the final closed cycle, therefore a(6) = 2.

Cf. A353311, A353312, A353313.

A353313(n) = { my(r=(n%3)); if(!r,n/3,((5*((n-r)/3)) + r + 3)); };
A353310(n) = { my(visited = Map(), p); for(j=0, oo, if(mapisdefined(visited, n, &p), return(p), mapput(visited, n, j)); n = A353313(n)); };

a(n) = A353311(n) - A353312(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).

A353305 The smallest number reached after n when iterating map x -> A353313(x) and starting from x=n, or -1 if no finite cycle is ever reached.

Original entry on oeis.org

0, 1, 5, 1, 1, 5, 2, 14, 5, 1, 5, 20, 1, 5, 25, 5, 16, 5, 2, 5, 20, 7, 5, 23, 5, 25, 5, 1, 28, 16, 5, 2, 5, 11, 5, 20, 1, 11, 5, 5, 23, 20, 14, 20, 25, 5, 25, 5, 16, 28, 16, 5, 16, 5, 2, 5, 16, 5, 11, 5, 20, 5, 20, 7, 11, 5, 5, 5, 16, 23, 20, 23, 5, 5, 20, 25, 20, 5, 5, 25, 5, 1, 5, 5, 28, 16, 20, 16, 20, 16, 5, 5

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, thus after the initial 2, 5 is the smallest number encountered, therefore a(2) = 5.

Cf. A353306 (positions of 1's), A353313.

A353313(n) = { my(r=(n%3)); if(!r,n/3,((5*((n-r)/3)) + r + 3)); };
A353305(n) = { my(visited = Map(), m=0); for(j=1, oo, n = A353313(n); if(!m, m=n, m=min(m,n)); if(mapisdefined(visited, n), return(m), mapput(visited, n, j))); };

A353309 The maximum sum of base-3 digits occurring among all numbers reached after n, when iterating map x -> A353313(x) starting from x=n, or -1 if no finite cycle is ever reached.

Original entry on oeis.org

0, 2, 18, 2, 2, 18, 18, 6, 18, 2, 18, 5, 2, 18, 6, 18, 6, 18, 18, 18, 5, 6, 18, 5, 18, 6, 18, 2, 5, 6, 18, 18, 18, 5, 18, 5, 2, 6, 18, 18, 5, 13, 6, 13, 6, 18, 8, 18, 6, 5, 6, 18, 6, 18, 18, 18, 8, 18, 5, 18, 5, 18, 5, 6, 6, 18, 18, 18, 8, 5, 13, 5, 18, 18, 13, 6, 13, 18, 18, 8, 18, 2, 18, 18, 5, 6, 13, 6, 13, 6

Offset: 0

Antti Karttunen, Apr 13 2022

			When starting iterating A353313 from n=7, we obtain -> 14 -> 25 -> 44 -> 75 -> 25 -> 44 -> 75 -> 25 -> etc, ad infinitum. Applying A053735 to all distinct terms encountered after 7, that is [14, 25, 44, 75] gives us base-3 digit sums [4, 5, 6, 5], therefore a(7) = 6, which is the largest sum.

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

Cf. A053735, A353313.

Cf. also A352895.

A053735(n) = sumdigits(n, 3);
A353313(n) = { my(r=(n%3)); if(!r,n/3,((5*((n-r)/3)) + r + 3)); };
A353309(n) = { my(visited = Map(), m=0); for(j=1, oo, n = A353313(n); m=max(m,A053735(n)); if(mapisdefined(visited, n), return(m), mapput(visited, n, j))); };

A353306 Numbers k such that 1 is in the transitive closure of the map x -> A353313(x) when starting iterating from x=k.

Original entry on oeis.org

1, 3, 4, 9, 12, 27, 36, 81, 108, 193, 243, 324, 346, 436, 579, 729, 972, 1038, 1308, 1522, 1737, 1867, 2187, 2353, 2539, 2916, 3114, 3493, 3924, 4234, 4566, 5211, 5601, 5824, 6286, 6561, 7059, 7617, 8748, 9342, 9446, 9709, 10479, 10886, 11756, 11772, 12702, 13698, 13772, 14792, 14806, 15633, 15745, 16184, 16803, 17003

Offset: 1

Antti Karttunen, Apr 13 2022

nonn

Positions of ones in A353305.

Subsequences: A000244, A003946, A211221.

A353313(n) = { my(r=(n%3)); if(!r,n/3,((5*((n-r)/3)) + r + 3)); };
A353305(n) = { my(visited = Map(), m=0); for(j=1, oo, n = A353313(n); if(!m, m=n, m=min(m,n)); if(mapisdefined(visited, n), return(m), mapput(visited, n, j))); };
isA353306(n) = (1==A353305(n));

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.

Original entry on oeis.org

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

A349877 a(n) is the number of times the map x -> A353314(x) needs to be applied to n to reach a multiple of 3, or -1 if the trajectory never reaches a multiple of 3.

Original entry on oeis.org

0, 2, 14, 0, 1, 13, 0, 4, 1, 0, 12, 3, 0, 1, 3, 0, 4, 1, 0, 11, 2, 0, 1, 2, 0, 2, 1, 0, 2, 3, 0, 1, 3, 0, 10, 1, 0, 4, 5, 0, 1, 7, 0, 3, 1, 0, 3, 2, 0, 1, 2, 0, 2, 1, 0, 2, 4, 0, 1, 9, 0, 3, 1, 0, 3, 4, 0, 1, 5, 0, 6, 1, 0, 4, 2, 0, 1, 2, 0, 2, 1, 0, 2, 7, 0, 1, 4, 0, 6, 1, 0, 6, 3, 0, 1, 3, 0, 5, 1, 0, 8, 2, 0

Offset: 0

Nicholas Drozd, Dec 03 2021

Equally, number of iterations of A353313 needed to reach a multiple of 3, or -1 if no multiple of 3 is ever reached. - Antti Karttunen, Apr 14 2022

			a(1) = 2 : 1 -> 4 -> 9 (as it takes two applications of A353314 to reach a multiple of three),
a(2) = 14 : 2 -> 5 -> 10 -> 19 -> 34 -> 59 -> 100 -> 169 -> 284 -> 475 -> 794 -> 1325 -> 2210 -> 3685 -> 6144
a(3) = 0 : 3 (as the starting point 3 is already a multiple of 3).
a(4) = 1 : 4 -> 9
a(7) = 4 : 7 -> 14 -> 25 -> 44 -> 75.

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

Cf. A010872, A349876, A353311, A353313.

A353314(n) = { my(r=(n%3)); if(!r,n,((5*((n-r)/3)) + r + 3)); };
A349877(n) = { my(k=0); while(n%3, k++; n = A353314(n)); (k); }; \\ Antti Karttunen, Apr 14 2022

import itertools
def f(n):
    for i in itertools.count():
        quot, rem = divmod(n, 3)
        if rem == 0:
            return i
        n = (5 * quot) + rem + 3

From Antti Karttunen, Apr 14 2022: (Start)
If A010872(n) = 0 then a(n) = 0, otherwise a(n) = 1 + a(A353314(n)).
a(n) < A353311(n) for all n.
(End)

Definition corrected and more terms from Antti Karttunen, Apr 14 2022

cp's OEIS Frontend

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

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A353310 Number of terms encountered when iterating A353313, before reaching the first term that is a part of a finite cycle, or -1 if no finite cycle is ever reached.

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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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A353305 The smallest number reached after n when iterating map x -> A353313(x) and starting from x=n, or -1 if no finite cycle is ever reached.

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A353309 The maximum sum of base-3 digits occurring among all numbers reached after n, when iterating map x -> A353313(x) starting from x=n, or -1 if no finite cycle is ever reached.

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A353306 Numbers k such that 1 is in the transitive closure of the map x -> A353313(x) when starting iterating from x=k.

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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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A349877 a(n) is the number of times the map x -> A353314(x) needs to be applied to n to reach a multiple of 3, or -1 if the trajectory never reaches a multiple of 3.

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