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A347755 Least k that does not appear in A347113(m), 1 <= m <= n.

%I A347755 #15 Nov 08 2021 11:58:50
%S A347755 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 A347755 17,17,17,17,17,17,17,17,19,19,19,19,19,19,19,19,19,19,19,19,19,19,23,
%U A347755 23,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31
%N A347755 Least k that does not appear in A347113(m), 1 <= m <= n.
%C A347755 a(0) = 1 by definition, since A347113 = 1 by definition of that sequence.
%C A347755 Lower bound on A347113.
%C A347755 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 A347755 Michael De Vlieger, <a href="/A347755/b347755.txt">Table of n, a(n) for n = 0..10000</a>
%H A347755 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 A347755 Let b(n) = A347113(n).
%e A347755 a(1) = 2 since b(1) = a(0) = 1.
%e A347755 a(k) = 2 for 1 <= k <= 7 since b(k) > 2.
%e A347755 a(8) = 3 since b(8) = a(7) = 2.
%e A347755 a(k) = 3 for 9 <= k <= 10 since b(k) > 3.
%e A347755 a(11) = 7 since b(11) = a(10) = 3.
%e A347755 a(k) = 7 for 12 <= k <= 17 since b(k) > 7, etc.
%t A347755 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 A347755 (* or using A347113 bfile: *)
%t A347755 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 A347755 (Python)
%o A347755 from math import gcd
%o A347755 A347755_list, nset, m, j = [1], {1}, 2, 2
%o A347755 for _ in range(10**2):
%o A347755     k = m
%o A347755     while k == j or gcd(k,j) == 1 or k in nset:
%o A347755         k += 1
%o A347755     j = k + 1
%o A347755     nset.add(k)
%o A347755     A347755_list.append(m)
%o A347755     while m in nset:
%o A347755         m += 1 # _Chai Wah Wu_, Sep 13 2021
%Y A347755 Cf. A008578, A347113, A347307, A347756 (distinct terms in this sequence).
%K A347755 nonn
%O A347755 0,2
%A A347755 _Michael De Vlieger_, Sep 12 2021