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Least k that does not appear in A347113(m), 1 <= m <= n.
3

%I #15 Nov 08 2021 11:58:50

%S 1,2,2,2,2,2,2,2,3,3,3,7,7,7,7,7,7,7,7,7,11,11,11,11,11,11,11,11,17,

%T 17,17,17,17,17,17,17,17,19,19,19,19,19,19,19,19,19,19,19,19,19,19,23,

%U 23,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31

%N Least k that does not appear in A347113(m), 1 <= m <= n.

%C a(0) = 1 by definition, since A347113 = 1 by definition of that sequence.

%C Lower bound on A347113.

%C Conjecture: all terms are in A008578. This is true for n <= 327680. Let j = A347113(m-1) and k = A347113(m) for k in A347757. For m > 0, k | j.

%H Michael De Vlieger, <a href="/A347755/b347755.txt">Table of n, a(n) for n = 0..10000</a>

%H Michael De Vlieger, <a href="/A347755/a347755.png">Log-log scatterplot of A347113(n)</a> for 1 <= n <= 2^12, showing terms in A347307 in red, those in this sequence joined in gold, and local minima in A347756 in blue

%e Let b(n) = A347113(n).

%e a(1) = 2 since b(1) = a(0) = 1.

%e a(k) = 2 for 1 <= k <= 7 since b(k) > 2.

%e a(8) = 3 since b(8) = a(7) = 2.

%e a(k) = 3 for 9 <= k <= 10 since b(k) > 3.

%e a(11) = 7 since b(11) = a(10) = 3.

%e a(k) = 7 for 12 <= k <= 17 since b(k) > 7, etc.

%t Block[{nn = 71, a = {1}, c, k, m, u = 2, v}, v = a; Map[Set[c[#], 1] &, Union@ a]; Do[Set[k, u]; If[PrimeQ[#], m = 2; While[IntegerQ[c[m #]], m++]; k = m #, While[Or[IntegerQ[c[k]], k == #, GCD[k, #] == 1], k++]] &[a[[-1]] + 1]; AppendTo[a, k]; Set[c[k], 1]; AppendTo[v, u]; If[k == u, While[IntegerQ[c[u]], u++]], nn]; v]

%t (* or using A347113 bfile: *)

%t Block[{a, u = {1}, v = 1}, a = Import["https://oeis.org/A347113/b347113.txt", "Data"][[All, -1]]; Do[If[a[[i]] == v, While[! FreeQ[a[[1 ;; i]], v], v++]]; AppendTo[u, v], {i, Length[a]}]; u]

%o (Python)

%o from math import gcd

%o A347755_list, nset, m, j = [1], {1}, 2, 2

%o for _ in range(10**2):

%o k = m

%o while k == j or gcd(k,j) == 1 or k in nset:

%o k += 1

%o j = k + 1

%o nset.add(k)

%o A347755_list.append(m)

%o while m in nset:

%o m += 1 # _Chai Wah Wu_, Sep 13 2021

%Y Cf. A008578, A347113, A347307, A347756 (distinct terms in this sequence).

%K nonn

%O 0,2

%A _Michael De Vlieger_, Sep 12 2021