A347757 -id:A347757 - cp's OEIS frontend

A347756 Local minima in A347113.

1, 2, 3, 7, 11, 17, 19, 23, 31, 37, 59, 61, 67, 79, 97, 151, 157, 199, 211, 229, 271, 307, 337, 367, 499, 577, 601, 619, 691, 727, 829, 877, 937, 1009, 1237, 1279, 1297, 1399, 1459, 1531, 1609, 1657, 1867, 2011, 2029, 2089, 2131, 2137, 2179, 2281, 2311, 2467, 2539

Offset: 1

Michael De Vlieger, Sep 12 2021

nonn

Distinct terms in A347755.

Conjecture: subset of A008578.

Chai Wah Wu, Table of n, a(n) for n = 1..6626 (terms n = 1..899 from Michael De Vlieger)
Michael De Vlieger, Log-log scatterplot of A347113(n) for 1 <= n <= 2^12, showing terms in A347307 in red, those in A347755 joined in gold, and terms in this sequence in blue.

Cf. A008578, A347113, A347307, A347755, A347757.

Block[{nn = 2^13, 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]; Union@ 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]}];  Union@ u]

from math import gcd
A347756_list, nset, m, j = [1], {1}, 2, 2
for _ in range(10**4):
    k = m
    while k == j or gcd(k,j) == 1 or k in nset:
        k += 1
    j = k + 1
    nset.add(k)
    if k == m:
        A347756_list.append(k)
    while m in nset:
        m += 1 # Chai Wah Wu, Sep 13 2021

A347757(n) = index of a(n) in A347113.

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

Original entry on oeis.org

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

Michael De Vlieger, Sep 12 2021

nonn

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.

			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.

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

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

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]

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

cp's OEIS Frontend

A347756 Local minima in A347113.

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