A347307 -id:A347307 - cp's OEIS frontend

A347113 a(1)=1; for n > 1, a(n) is the smallest unused positive number k such that k != j and gcd(k,j) != 1, where j = a(n-1) + 1.

1, 4, 10, 22, 46, 94, 5, 2, 6, 14, 3, 8, 12, 26, 9, 15, 18, 38, 13, 7, 16, 34, 20, 24, 30, 62, 21, 11, 27, 32, 36, 74, 25, 28, 58, 118, 17, 33, 40, 82, 166, 334, 35, 39, 42, 86, 29, 44, 48, 56, 19, 45, 23, 50, 54, 60, 122, 41, 49, 52, 106, 214, 43, 55, 63, 66

Offset: 1

Grant Olson, Aug 18 2021

nonn

Alternative definition: Lexicographically earliest sequence of distinct positive numbers such that a(n) != a(n-1)+1 and gcd(a(n-1)+1,a(n)) > 1. This makes it a cousin of the EKG sequence A064413, the Yellowstone permutation A098550, the Enots Wolley sequence A336957, and others. - N. J. A. Sloane, Sep 01 2021; revised Nov 08 2021.

The successive gcd's are listed in A347309.

			a(1) = 1, by definition.
a(2) = 4; it cannot be 2, because 2 = a(1) + 1, and it cannot be 3, because gcd(a(1) + 1, 3) = 1.
a(3) = 10, because gcd(a(3), a(2) + 1) cannot equal 1. a(2) + 1 = 5, so a(3) must be a multiple of 5. It cannot be equal to 5, so it must be 10, the next available multiple of 5.
a(4) = 22, because 22 is the smallest positive integer not equal to 11 and not coprime to 11.

Michel Marcus, Table of n, a(n) for n = 1..10000
Chai Wah Wu, Table of n, a(n) for n = 1..10^6
Chai Wah Wu, Graph of first million terms
Index entries for sequences that are permutations of the integers

See A347306 for the inverse, A347307, A347308 for the records, A347309 for the gcd values, A347312 for the parity of a(n), A347314 for the fixed points, and A348780 for partial sums.
For the main diagonal see (A348787(k), A348788(k)).
Cf. A064413, A098550, A336957, A347313, A348779, A348785, A348786, A353712, A353713.

b:= proc() true end:
a:= proc(n) option remember; local j, k; j:= a(n-1)+1;
      for k from 2 do if b(k) and k<>j and igcd(k, j)>1
        then b(k):= false; return k fi od
    end: a(1):= 1:
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 02 2021

Block[{a = {1}, c, k, m = 2}, Do[If[IntegerQ@Log2[i], While[IntegerQ[c[m]], m++]]; Set[k, m]; While[Or[IntegerQ[c[k]], k == # + 1, GCD[k, # + 1] == 1], k++] &[a[[-1]]]; AppendTo[a, k]; Set[c[k], i], {i, 65}]; a] (* Michael De Vlieger, Aug 18 2021 *)

find(va, x) = {my(k=1, s=Set(va)); while ((k==x) || (gcd(k, x) == 1) || setsearch(s, k), k++); k;}
lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, va[n] = find(va, va[n-1]+1);); va;} \\ Michel Marcus, Aug 21 2021

from math import gcd
A347113_list, nset, m = [1], {1}, 2
for _ in range(100):
    j = A347113_list[-1]+1
    k = m
    while k == j or gcd(k,j) == 1 or k in nset:
        k += 1
    A347113_list.append(k)
    nset.add(k)
    while m in nset:
        m += 1 # Chai Wah Wu, Sep 01 2021

Comments edited (including deletion of incorrect comments) by N. J. A. Sloane, Sep 05 2021

For the moment I am withdrawing my claim that this is a permutation of the positive integers. - N. J. A. Sloane, Sep 05 2022

A347756 Local minima in A347113.

Original entry on oeis.org

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.

A347308 Indices of records in A347113.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 36, 41, 42, 90, 91, 92, 93, 94, 519, 1044, 1251, 1252, 1422, 2748, 3591, 6528, 10685, 11661, 12028, 12236, 17326, 19899, 20074, 22571, 26429, 27702, 30538, 43975, 54016, 54017, 54229, 61703, 63862, 63863, 127935, 127936, 269513, 297679, 342675

Offset: 1

N. J. A. Sloane, Sep 01 2021

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..54

Cf. A347113, A347306, A347307.

from math import gcd
A347308_list, nset, m, c, j, i = [1], {1}, 2, 0, 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
    if k > c:
        c = k
        A347308_list.append(i)
    j = k + 1
    nset.add(k)
    while m in nset:
        m += 1 # Chai Wah Wu, Sep 01 2021

a(23)-a(40) from Alois P. Heinz, Sep 01 2021
a(41)-a(42) from Chai Wah Wu, Sep 01 2021
a(43)-a(45) from Chai Wah Wu, Sep 02 2021

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

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

A347113 a(1)=1; for n > 1, a(n) is the smallest unused positive number k such that k != j and gcd(k,j) != 1, where j = a(n-1) + 1.

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A347756 Local minima in A347113.

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A347308 Indices of records in A347113.

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

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

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

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