A349876 -id:A349876 - cp's OEIS frontend

A349896 Record values in A349876.

0, 9, 6144, 8850, 63294, 167460, 1471350, 17358894, 19273044, 90701559, 153178644, 189719685, 197747394, 2017743225, 6233637435, 59571334269, 86021383575, 156569713710, 2073505928019, 2889765691185, 17962980751950, 56300638961700, 277087084821075, 329363647943184

Offset: 1

Nicholas Drozd, Dec 04 2021

Cf. A349876.

f(n) = my(d=divrem(n, 3)); if (d[2], f(5*d[1]+d[2]+3), n); \\ A349876
lista(nn) = {my(r=0); for (n=1, nn, my(x = f(n)); if (x>r, print1(x, ", "); r=x););} \\ Michel Marcus, Dec 06 2021

A353313 If n is of the form 3k, then a(n) = k, 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, 1, 9, 10, 2, 14, 15, 3, 19, 20, 4, 24, 25, 5, 29, 30, 6, 34, 35, 7, 39, 40, 8, 44, 45, 9, 49, 50, 10, 54, 55, 11, 59, 60, 12, 64, 65, 13, 69, 70, 14, 74, 75, 15, 79, 80, 16, 84, 85, 17, 89, 90, 18, 94, 95, 19, 99, 100, 20, 104, 105, 21, 109, 110, 22, 114, 115, 23, 119, 120, 24, 124, 125, 25, 129, 130, 26

Offset: 0

Antti Karttunen, Apr 13 2022

nonn

It is conjectured that all iterations of this sequence starting from any n >= 0 will eventually reach a finite cycle, which by necessity then contains at least one multiple of three. See Drozd links and A349876.

Antti Karttunen, Table of n, a(n) for n = 0..19683
Nicholas Drozd, A Busy Beaver Champion Derived from Scratch
Nicholas Drozd, Feedback to Doron Zeilberger's opinion #155, Jan. 4, 2022

Cf. A349876, A353310, A353311, A353312, A353314 (variant).

Cf. A353305 (the smallest number reached after the starting point n), A353309 (the largest base-3 digit sum reached after the starting point n).

Table[With[{c=Mod[n,3]},If[c==0,n/3,(5n-2c+9)/3]],{n,0,80}]  (* Harvey P. Dale, Aug 09 2025 *)

A353313(n) = { my(r=(n%3)); if(!r,n/3,((5*((n-r)/3)) + r + 3)); };

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)

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

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A349896 Record values in A349876.

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A353313 If n is of the form 3k, then a(n) = k, 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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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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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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