A347756 -id:A347756 - cp's OEIS frontend

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

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

A347757 Indices of local minima in A347113.

Original entry on oeis.org

1, 8, 11, 20, 28, 37, 51, 53, 101, 116, 136, 146, 159, 213, 302, 318, 440, 520, 638, 698, 702, 912, 1031, 1128, 1528, 1758, 1832, 1891, 2107, 2198, 2523, 2671, 2857, 3069, 3760, 3892, 3946, 4256, 4438, 4638, 4880, 5022, 5656, 6092, 6147, 6322, 6470, 6492, 6579

Offset: 1

Michael De Vlieger, Sep 12 2021

nonn

a(n)-1 = last instance of A347756(n) in A347755.

a(n+1) > a(n) + 1, since terms in A347113 are distinct by definition.

Chai Wah Wu, Table of n, a(n) for n = 1..6904 (terms 1..898 from Michael De Vlieger)

Cf. A347113, A347306 (indices of records in A347113), A347755, A347756.

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]; Map[FirstPosition[a, #][[1]] &, Most@ 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]}]; Map[FirstPosition[a, #][[1]] &, Most@ Union@ u] ]

from math import gcd
A347757_list, nset, m, j, i = [1], {1}, 2, 2, 1
for _ in range(10**4):
    i += 1
    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:
        A347757_list.append(i)
    while m in nset:
        m += 1 # Chai Wah Wu, Sep 13 2021

A349405 a(n) = A347113(A347313(n))+1.

Original entry on oeis.org

95, 6, 15, 39, 14, 22, 119, 87, 57, 46, 123, 215, 159, 94, 93, 219, 74, 118, 122, 303, 142, 134, 327, 166, 695, 178, 395, 206, 214, 226, 447, 959, 262, 254, 543, 291, 302, 326, 334, 699, 346, 358, 382, 386, 394, 843, 1727, 879, 446, 454, 478, 482, 502, 8159, 514

Offset: 1

Michael De Vlieger, Nov 16 2021

nonn

These numbers generate primes in A347113.
Let s = A347113, j = s(i)+1, and k = s(i+1). For prime k, j is a squarefree semiprime pq, p < q.
The first 3 primes in s have k = p, while all others observed for i <= 2^19 have k = q.

			a(1) = s(6)+1 = 95 -> s(7) = 5,
a(2) = s(7)+1 = 6 -> s(8) = 2,
a(3) = s(10)+1 = 15, -> s(11) = 3,
a(4) = s(18)+1 = 39, -> s(19) = 13, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Extended table of n, a(n) for n = 1..23082 (all terms resulting from 2^19 terms of A347113)
Michael De Vlieger, Log-log scatterplot of A347113(n) n=1..256, indicating primes in green, labeling them with the value a(n) = A347113(n-1)+1. Red dots represent records A347307 in A347113, and blue represent local minima A347756 in A347113.

Cf. A006530, A020639, A347113, A347313, A348779, A349406.

c[] = 0; j = m = 2; m = {1}~Join~Reap[Do[If[IntegerQ@ Log2[i], While[c[m] > 0, m++]]; Set[k, m]; While[Or[c[k] > 0, k == j, GCD[j, k] == 1], k++]; Sow[k]; Set[c[k], i]; j = k + 1, {i, 530}]][[-1, -1]]; m[[Position[m, ?PrimeQ][[All, 1]] - 1]] + 1

A349406 a(n) = A349405(n)/A348779(n).

Original entry on oeis.org

19, 3, 5, 3, 2, 2, 7, 3, 3, 2, 3, 5, 3, 2, 3, 3, 2, 2, 2, 3, 2, 2, 3, 2, 5, 2, 5, 2, 2, 2, 3, 7, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 11, 3, 2, 2, 2, 2, 2, 41, 2, 2, 3, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 5, 3, 5, 3, 3, 5, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 3, 2, 2

Offset: 1

Michael De Vlieger, Nov 16 2021

nonn

Ratio of progenitor and prime in A347113.
Let s = A347113, j = s(i)+1, and k = s(i+1). For prime k, j is a squarefree semiprime pq, p < q.
The first 3 primes in s have k = p, while all others observed for i <= 2^19 have k = q. This sequence thus lists the other prime factor r of j such that r*k = j.
The quasi-linear striations k < n are arranged according to this sequence (see color-coded log-log scatterplot). - Michael De Vlieger, Nov 17 2021

			s(6)+1 = 95 -> s(7) = 5; a(1) = 95/5 = 19.
s(7)+1 = 6 -> s(8) = 2; a(2) = 6/2 = 3.
s(10)+1 = 15, -> s(11) = 3; a(3) = 15/3 = 5.
s(18)+1 = 39, -> s(19) = 13; a(4) = 39/13 = 3, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Notes on ratio j/k in A347113, listing numbers of certain ratios seen given 2^19 terms of A347113.
Michael De Vlieger, Extended table of n, a(n) for n = 1..23082 (all terms resulting from 2^19 terms of A347113)
Michael De Vlieger, Log-log scatterplot of A347113(n), n=1..256 indicating primes in green, labeling them with the ratio j/k = a(n) = (A347113(n-1)+1)/A347113(n). Red dots represent records A347307 in A347113, and blue represent local minima A347756 in A347113.
Michael De Vlieger, Log-log scatterplot of a(n), for n=1..2^16 showing primes k=A347113(A348784) listed in A348779, color coded according to the ratio j/k, which is always prime. Color function: red = 2, orange = 3, chartreuse = 5, small green = 7, large green = 11, large cyan = 13, large blue = 17, large indigo = 19, large purple = 29, large magenta = 41.

Cf. A006530, A020639, A347113, A347313, A348779, A349405.

c[] = 0; j = m = 2; m = {1}~Join~Reap[Do[If[IntegerQ@ Log2[i], While[c[m] > 0, m++]]; Set[k, m]; While[Or[c[k] > 0, k == j, GCD[j, k] == 1], k++]; Sow[k]; Set[c[k], i]; j = k + 1, {i, 900}]][[-1, -1]]; Map[(#1 + 1)/#2 & @@ m[[# - 1 ;; #]] &, Position[m, ?PrimeQ][[All, 1]]]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

A347757 Indices of local minima in A347113.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

A349405 a(n) = A347113(A347313(n))+1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A349406 a(n) = A349405(n)/A348779(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica