A364447 -id:A364447 - cp's OEIS frontend

A364448 Lexicographically earliest sequence where n is banned for n^2 terms after each appearance.

1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 1, 6, 1, 2, 1, 3, 1, 7, 1, 2, 1, 8, 1, 4, 1, 2, 1, 3, 1, 9, 1, 2, 1, 10, 1, 5, 1, 2, 1, 3, 1, 4, 1, 2, 1, 11, 1, 12, 1, 2, 1, 3, 1, 6, 1, 2, 1, 13, 1, 4, 1, 2, 1, 3, 1, 5, 1, 2, 1, 7, 1, 14, 1, 2, 1, 3, 1, 4, 1, 2, 1, 15, 1, 16, 1, 2, 1, 3, 1, 8, 1, 2, 1, 5, 1, 4

Offset: 1

Rok Cestnik, Jul 25 2023

After 12427560 terms the sequence is periodic with period 2985984. The largest periodic term is 158.
The largest term is 159 which appears only 3 times: a(4942686), a(7756992) and a(9818928).
If banning for n terms the sequence just repeats [1,2,1,3] (A364447). If banning for n^3 terms the sequence continues growing forever (A364449).
The first 41 terms are shared with A028920.

			a(n)   ban 1  2  3  4  5  6  7  ...
 1         |  |  |  |  |  |  |
 2         x  |  |  |  |  |  |
 1         |  x  |  |  |  |  |
 3         x  x  |  |  |  |  |
 1         |  x  x  |  |  |  |
 4         x  x  x  |  |  |  |
 1         |  |  x  x  |  |  |
 2         x  |  x  x  |  |  |
 1         |  x  x  x  |  |  |
 5         x  x  x  x  |  |  |
 1         |  x  x  x  x  |  |
 6         x  x  x  x  x  |  |
 1         |  |  x  x  x  x  |
 2         x  |  |  x  x  x  |
 1         |  x  |  x  x  x  |
 3         x  x  |  x  x  x  |
 1         |  x  x  x  x  x  |
 7         x  x  x  x  x  x  |
 .
 .
 .

Rok Cestnik, Table of n, a(n) for n = 1..10000

Cf. A364447, A364449, A028920.

#include
int main(void){
    int N = 1000;
    int *a = (int*)malloc((N+1)*sizeof(int));
    int *ban = (int*)malloc(160*sizeof(int));
    for(int n = 0; n < N; ++n){
        for(int can = 1; can < 160; ++can){
            if(ban[can] == 0){
                a[n] = can;
                ban[can] = can*can+1;
                break;
            }
        }
        for(int can = 1; can < 160; ++can) if(ban[can]) --ban[can];
    }
    free(a); free(ban);
    return 0;
}

a = []
ban = [0 for n in range(160)]
for i in range(1000):
    can = ban.index(0,1)
    ban = [max(b-1,0) for b in ban]
    a.append(can)
    ban[can] = can**2

a(n) = a(n-2985984) for n >= 15413545.

A364449 Lexicographically earliest sequence where n is banned for n^3 terms after each appearance.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 2, 1, 7, 1, 8, 1, 9, 1, 10, 1, 2, 1, 11, 1, 12, 1, 13, 1, 14, 1, 2, 1, 3, 1, 15, 1, 16, 1, 17, 1, 2, 1, 18, 1, 19, 1, 20, 1, 21, 1, 2, 1, 22, 1, 23, 1, 24, 1, 25, 1, 2, 1, 3, 1, 26, 1, 27, 1, 28, 1, 2, 1, 4, 1, 29, 1, 30, 1, 31, 1, 2, 1, 32, 1, 33, 1, 34, 1, 35

Offset: 1

Rok Cestnik, Jul 25 2023

nonn

Sequence is unbounded. The fastest branch grows asymptotically linearly: limsup a(n)/n > 1-Sum_{n>0} 1/(n^3+1) = 1-A339606 = 0.313496...

If banning for n terms (A364447), or n^2 terms (A364448), the sequence is eventually periodic.

			a(n)   ban 1  2  3  4  5  6  7  ...
 1         |  |  |  |  |  |  |
 2         x  |  |  |  |  |  |
 1         |  x  |  |  |  |  |
 3         x  x  |  |  |  |  |
 1         |  x  x  |  |  |  |
 4         x  x  x  |  |  |  |
 1         |  x  x  x  |  |  |
 5         x  x  x  x  |  |  |
 1         |  x  x  x  x  |  |
 6         x  x  x  x  x  |  |
 1         |  |  x  x  x  x  |
 2         x  |  x  x  x  x  |
 1         |  x  x  x  x  x  |
 7         x  x  x  x  x  x  |
 1         |  x  x  x  x  x  x
 .
 .
 .

Cf. A364447, A364448, A339606.

a = []
ban = [0 for n in range(500)]
for i in range(1000):
    can = ban.index(0,1)
    ban = [max(b-1,0) for b in ban]
    a.append(can)
    ban[can] = can**3

cp's OEIS Frontend

A364448 Lexicographically earliest sequence where n is banned for n^2 terms after each appearance.

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