A347113 -id:A347113 - cp's OEIS frontend

A347306 When k appears in A347113, or -1 if k never appears.

1, 8, 11, 2, 7, 9, 20, 12, 15, 3, 28, 13, 19, 10, 16, 21, 37, 17, 51, 23, 27, 4, 53, 24, 33, 14, 29, 34, 47, 25, 101, 30, 38, 22, 43, 31, 116, 18, 44, 39, 58, 45, 63, 48, 52, 5, 99, 49, 59, 54, 68, 60, 81, 55, 64, 50, 73, 35, 136, 56, 146, 26, 65, 69, 72, 66, 159, 74, 77, 70, 153

Offset: 1

N. J. A. Sloane, Sep 01 2021

nonn

It is conjectured that every positive number appears in A347113. - N. J. A. Sloane, Nov 08 2021

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 3330 terms from N. J. A. Sloane)
Hugo Pfoertner, Table of n, a(n) for n = 1..132708
Hugo Pfoertner, Illustration of terms corresponding to first 400000 terms of A347113, use zoom to see details.

Cf. A347113.

With[{s = Import["https://oeis.org/A347113/b347113.txt", "Data"][[All, -1]]}, Array[FirstPosition[s, #][[1]] &, 71]] (* Michael De Vlieger, Sep 01 2021 *)

from math import gcd
def A347306(n):
    if n == 1:
        return 1
    i, j, nset, m = 1, 2, {1}, 2
    while True:
        k = m
        i += 1
        while k == j or gcd(k,j) == 1 or k in nset:
            k += 1
        if k == n:
            return i
        j = k+1
        nset.add(k)
        while m in nset:
            m += 1 # Chai Wah Wu, Sep 02 2021

A347307 Records in A347113.

Original entry on oeis.org

1, 4, 10, 22, 46, 94, 118, 166, 334, 358, 718, 1438, 2878, 5758, 8158, 8254, 9838, 19678, 22558, 43198, 56638, 103198, 169438, 184798, 190558, 193918, 274558, 315358, 318238, 357598, 419038, 439678, 486238, 698398, 858238, 1716478, 1723198, 1965118, 2029438, 4058878

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, A347308.

from math import gcd
A347307_list, nset, m, c, j = [1], {1}, 2, 0, 2
for _ in range(10**4):
    k = m
    while k == j or gcd(k,j) == 1 or k in nset:
        k += 1
    if k > c:
        c = k
        A347307_list.append(k)
    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

A347313 Index of prime(n) in A347113, or -1 if that prime never appears.

Original entry on oeis.org

8, 11, 7, 20, 28, 19, 37, 51, 53, 47, 101, 116, 58, 63, 99, 81, 136, 146, 159, 153, 115, 213, 176, 197, 302, 151, 215, 223, 169, 230, 276, 274, 255, 188, 233, 318, 440, 341, 347, 359, 369, 282, 386, 396, 405, 520, 638, 460, 472, 698, 357, 492, 507, 514, 529, 535, 558, 702

Offset: 1

N. J. A. Sloane, Sep 06 2021

nonn

Conjecture: every prime appears in A347113 (every number, in fact).

The graph shows three three strong lines (and many other points). Can the primes on the three lines be described in a simple way?

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A347113.

from math import gcd
from sympy import prime
def A347313(n):
    p = prime(n)
    i, j, nset, m = 1, 2, {1}, 2
    while True:
        k = m
        i += 1
        while k == j or gcd(k,j) == 1 or k in nset:
            k += 1
        if k == p:
            return i
        j = k+1
        nset.add(k)
        while m in nset:
            m += 1 # Chai Wah Wu, Sep 06 2021

A348779 Primes in A347113 in order of appearance.

Original entry on oeis.org

5, 2, 3, 13, 7, 11, 17, 29, 19, 23, 41, 43, 53, 47, 31, 73, 37, 59, 61, 101, 71, 67, 109, 83, 139, 89, 79, 103, 107, 113, 149, 137, 131, 127, 181, 97, 151, 163, 167, 233, 173, 179, 191, 193, 197, 281, 157, 293, 223, 227, 239, 241, 251, 199, 257, 263, 349, 269, 283, 277, 401, 409, 311, 421, 211, 313

Offset: 1

N. J. A. Sloane, Nov 13 2021

nonn

From Michael De Vlieger, Nov 13 2021: (Start)
Let s = A347113, j = s(n-1)+1 and k = s(n). Prime k|j = q such that j/q = p, p < q, both primes, in all cases except the first 3, i.e., s(7), s(8), and s(11), with (j, k) = {(95, 5), (6, 2), (15, 3)} respectively.
In other words, squarefree semiprime j = pq, p < q, yields k = q outside of the first 3 primes in s. Are all prime s(n), n > 219 in this category?
Prime k implies k | j, since k = j is not permitted in s, k < j.
There is 1 instance of composite k | j, i.e., s(33) = 25, with j = 75. Are there any others?
The reverse relation j to k is that j is the product of at least one prime divisor p | k and at least one prime q that does not divide k. When k is prime p, j = pq.
Contains local minima in s aside from s(1). A consequence of forbidden j = k in s is that local minima are nonadjacent.
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..23082
Michael De Vlieger, Log-log scatterplot of A347113(n) for n=1..2^12 showing primes in this sequence in green. Maxima in A347113 appear in red and local minima in blue.

Cf. A347113.

c[] = 0; m = 1; k = 2; Reap[Monitor[Do[If[IntegerQ@ Log2[i], While[c[k] > 0, k++]]; Set[n, k]; While[Or[c[n] > 0, n == m + 1, GCD[n, m + 1] == 1], n++]; If[PrimeQ[n], Sow[n]]; Set[c[n], i]; m = n, {i, 648}], i]][[-1, -1]] (* _Michael De Vlieger, Nov 13 2021 *)

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

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

A348786 Trajectory of 5 under repeated application of the map x -> A347113(x).

Original entry on oeis.org

5, 46, 86, 77, 69, 64, 55, 54, 50, 56, 60, 52, 45, 42, 334, 335, 321, 297, 284, 267, 247, 237, 221, 203, 192, 382, 746, 1454, 1430, 2858, 2806, 1861, 1860, 1828, 1796, 1772, 1742, 1708, 1687, 1655, 1616, 1590, 1558, 1527, 1496, 1482, 1488, 1464, 1432, 1412, 1404, 1378, 1339, 1320, 1273, 1248

Offset: 1

N. J. A. Sloane, Nov 16 2021

nonn

Probably does not cycle, although the graph suggests that it could happen (compare A348785).

Chai Wah Wu, Table of n, a(n) for n = 1..274 (terms 1..232 from N. J. A. Sloane)

Cf. A347113, A347306, A348785.

A348787 Indices k such that (8/9)k < A347113(k) < (5/4)k.

Original entry on oeis.org

1, 13, 16, 17, 24, 25, 29, 30, 31, 39, 45, 48, 49, 50, 54, 55, 56, 65, 66, 69, 70, 72, 74, 75, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 95, 96, 97, 98, 100, 102, 105, 107, 108, 110, 112, 113, 117, 118, 119, 120, 121, 124, 125, 126, 129, 132, 133, 137, 138, 139, 140, 142, 143, 147, 148, 149, 152, 154, 155, 156, 160

Offset: 1

N. J. A. Sloane, Nov 20 2021

nonn

By definition, the points (a(k), A348788(k)) form the main diagonal of A347113. See A348788 for a discussion of this diagonal.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A347113, A348788.

cp's OEIS Frontend

A347306 When k appears in A347113, or -1 if k never appears.

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A347307 Records in A347113.

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A347313 Index of prime(n) in A347113, or -1 if that prime never appears.

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A348779 Primes in A347113 in order of appearance.

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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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A347757 Indices of local minima in A347113.

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A348786 Trajectory of 5 under repeated application of the map x -> A347113(x).

A348787 Indices k such that (8/9)k < A347113(k) < (5/4)k.