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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.

%I A364603 #24 Aug 14 2023 02:01:03
%S A364603 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,
%T A364603 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,
%U A364603 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
%N 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.
%H A364603 Rok Cestnik, <a href="/A364603/b364603.txt">Table of n, a(n) for n = 1..10000</a>
%H A364603 Michael De Vlieger, <a href="/A364603/a364603.png">Log log scatterplot of a(n)</a>, n = 1..2^16.
%H A364603 Michael De Vlieger, <a href="/A364603/a364603_1.png">Plot [b(k) > n] at (x,y) = (n,k)</a>, 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.
%e A364603 a(n)   ban 1  2  3  4  5  6  7 ...
%e A364603  1         |  |  |  |  |  |  |
%e A364603  2         x  |  |  |  |  |  |
%e A364603  1         |  x  |  |  |  |  |
%e A364603  3         x  x  |  |  |  |  |
%e A364603  2         x  |  x  |  |  |  |
%e A364603  1         |  x  x  |  |  |  |
%e A364603  4         x  x  x  |  |  |  |
%e A364603  3         x  x  |  x  |  |  |
%e A364603  5         x  x  x  x  |  |  |
%e A364603  1         |  |  x  x  x  |  |
%e A364603  2         x  |  x  x  x  |  |
%e A364603  4         x  x  x  |  x  |  |
%e A364603  6         x  x  x  x  x  |  |
%e A364603  7         x  x  x  x  x  x  |
%e A364603  1         |  x  |  x  |  x  x
%e A364603  3         x  x  |  x  |  x  x
%e A364603  5         x  x  x  x  |  x  x
%e A364603  2         x  |  x  x  x  x  x
%e A364603  8         x  x  x  x  x  x  x
%e A364603  6         x  x  x  x  x  |  x
%e A364603  1         |  x  x  |  x  x  x
%e A364603  .
%e A364603  .
%e A364603  .
%t A364603 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 *)
%o A364603 (PARI) 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}
%o A364603 (Python)
%o A364603 def A364603(N):
%o A364603     a = [0]*N; z=s=0
%o A364603     while(s<N):
%o A364603         z+=1; m=1; i=s
%o A364603         while(i<N):
%o A364603             if not a[i]:
%o A364603                 a[i]=z; i+=z*m; m+=1
%o A364603             i+=1
%o A364603         s+=1
%o A364603     return a
%Y A364603 Cf. A364604, A364448, A364449.
%K A364603 nonn,look
%O A364603 1,2
%A A364603 _Rok Cestnik_, Jul 29 2023