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

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, 17, 17, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 23, 23, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31

Offset: 0

Author

Michael De Vlieger, Sep 12 2021

Keywords

• nonn

Comments

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

Lower bound on A347113.

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.

Examples

			Let b(n) = A347113(n).
a(1) = 2 since b(1) = a(0) = 1.
a(k) = 2 for 1 <= k <= 7 since b(k) > 2.
a(8) = 3 since b(8) = a(7) = 2.
a(k) = 3 for 9 <= k <= 10 since b(k) > 3.
a(11) = 7 since b(11) = a(10) = 3.
a(k) = 7 for 12 <= k <= 17 since b(k) > 7, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Michael De Vlieger, Log-log scatterplot of A347113(n) 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

Crossrefs

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

Programs

Mathematica

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]
(* or using A347113 bfile: *)
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]

Python

from math import gcd
A347755_list, nset, m, j = [1], {1}, 2, 2
for _ in range(10**2):
    k = m
    while k == j or gcd(k,j) == 1 or k in nset:
        k += 1
    j = k + 1
    nset.add(k)
    A347755_list.append(m)
    while m in nset:
        m += 1 # Chai Wah Wu, Sep 13 2021