A364604 -id:A364604 - cp's OEIS frontend

A364603 Lexicographically earliest sequence where after the m-th appearance of term z, it is banned from re-appearing in the next m*z terms.

1, 2, 1, 3, 2, 1, 4, 3, 5, 1, 2, 4, 6, 7, 1, 3, 5, 2, 8, 6, 1, 4, 7, 9, 10, 3, 2, 1, 5, 8, 11, 12, 6, 9, 4, 1, 10, 2, 3, 7, 13, 14, 11, 15, 1, 5, 8, 12, 16, 17, 2, 4, 6, 9, 1, 3, 13, 10, 14, 15, 18, 7, 19, 20, 21, 1, 2, 5, 11, 16, 17, 8, 4, 12, 3, 22, 23, 1, 6, 18, 24, 9, 19, 2, 13, 20, 21, 14

Offset: 1

Rok Cestnik, Jul 29 2023

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

Rok Cestnik, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.
Michael De Vlieger, Plot [b(k) > n] at (x,y) = (n,k), for n = 1..2^12, where k may appear at index b(k) and brackets are Iverson. Hence if b(k) > n, k is banned and shown in black else k is permitted and shown in white. This image is similar to the Example but rotated 90 degrees counterclockwise.

Cf. A364604, A364448, A364449.

c[] := 0; q[] := 1; r = k = 1; nn = 120; Do[Set[{a[n], c[k]}, {k, n + k*q[k] + 1}]; q[k]++; If[k > r, r = k]; k = MinimalBy[Range[r], c, 1][[1]]; If[c[k] > n + 1, k = r + 1], {n, nn}]; Array[a, nn] (* Michael De Vlieger, Aug 13 2023 *)

A364603(N) = {my(a=vector(N),z=1); for(s=1,N, if(a[s],next); my(m=0); for(i=s,N, if(!a[i], a[i]=z; i+=(z*(m++)))); z++); a}

def A364603(N):
    a = [0]*N; z=s=0
    while(s

A372704 a(1)=1; a(2)=2; for n>1, this is the lexicographically earliest sequence such that, following each occurrence of a(n), a(n) is banned for the next k terms, where k is the number of terms prior to a(n) that are not equal to a(n).

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 2, 1, 4, 5, 6, 2, 1, 3, 7, 8, 9, 4, 10, 5, 2, 1, 6, 11, 12, 13, 3, 14, 15, 7, 16, 8, 17, 9, 4, 18, 19, 2, 1, 5, 10, 20, 21, 22, 6, 23, 24, 11, 25, 12, 26, 3, 13, 27, 28, 14, 29, 15, 7, 30, 31, 16, 8, 32, 33, 17, 9, 4, 34, 35, 2, 1, 18, 19, 36

Offset: 1

Neal Gersh Tolunsky, May 10 2024

nonn

1 occurs immediately after its ban ends so that the i-th occurrence of a(n) = 1, for i >= 2, is at n = 2^(i-2) + i = A052968(i-1).
2 occurs immediately after its ban ends, since it turns out that's immediately before the ban on 1 ends, so the i-th occurrence of a(n) = 2 is at n = 2^(i-1) + i = A005126(i-1).
If the definition is changed so that k is the number of terms in the sequence thus far equal to a(n) (rather than unequal), this becomes A002260 without the initial 1.

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

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Neal Gersh Tolunsky, Graph of first differences of first 20000 terms.

Cf. A364603, A364604.

from collections import Counter
from itertools import count, islice
def agen(): # generator of terms
    an, ban, occurs = 1, {1: 1}, Counter([1])
    for n in count(2):
        yield an
        an = next(k for k in count(1) if k not in ban)
        for k in list(ban):
            if ban[k] > 1: ban[k] -= 1
            else: del ban[k]
        ban[an] = n - 1 - occurs[an]
        occurs[an] += 1
print(list(islice(agen(), 75))) # Michael S. Branicky, May 10 2024

a(20) and beyond from Michael S. Branicky, May 10 2024

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A364603 Lexicographically earliest sequence where after the m-th appearance of term z, it is banned from re-appearing in the next m*z terms.

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