A347313 -id:A347313 - cp's OEIS frontend

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

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

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.

Original entry on oeis.org

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

A348777 a(n) = term in A347113 that precedes the n-th prime. Set a(n) = -1 if prime(n) does not appear in A347113.

Original entry on oeis.org

5, 14, 94, 13, 21, 38, 118, 56, 45, 86, 92, 73, 122, 214, 93, 158, 117, 121, 133, 141, 218, 394, 165, 177, 290, 302, 205, 213, 326, 225, 253, 261, 958, 694, 446, 301, 1726, 325, 333, 345, 357, 542, 381, 385, 393, 8158, 421, 445, 453, 686, 698, 477, 481, 501, 513, 525, 537, 1354, 553, 842

Offset: 1

N. J. A. Sloane, Nov 08 2021

nonn

			A347113(11) = 3 is preceded by A347113(10) = 14, and 3 is the 2nd prime, so a(2) = 14. Also (14+1)/3 = 5 = A348778(2).

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A347113, A347313, A348778.

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]]]

A349412 Positions of prime j = A347113(i)+1.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 13, 17, 21, 25, 31, 34, 35, 39, 40, 41, 45, 56, 60, 61, 66, 70, 75, 79, 88, 89, 90, 91, 92, 93, 102, 105, 108, 113, 121, 122, 126, 127, 133, 140, 143, 149, 156, 160, 167, 180, 185, 186, 191, 192, 199, 203, 208, 211, 231, 235, 240, 241, 245

Offset: 1

Michael De Vlieger, Nov 16 2021

nonn

Let s = A347113, j = s(i)+1 and k = s(i+1). We recall the 3 constraints presented in A347113:
1. j = k is forbidden.
2. gcd(j,k) = 1 is forbidden.
3. All terms in s are distinct.
These constraints confine prime j to the relationship j | k, which in the context of s, implies j < k and increase. The least k > j such that j | k is 2j, giving rise to Cunningham chains of the first kind. The chains are evident in this sequence as series of consecutive positions in s.

			s(1) = 1, thus j = s(1)+1 = 2, which is prime, therefore a(1) = 1.
s(2) = 4; j = 5, thus a(2) = 2, etc.

Chris Caldwell's Prime Glossary, Cunningham chains.
Michael De Vlieger, Extended table of n, a(n) for n = 1..45450
Eric Weisstein's World of Mathematics, Cunningham Chain.

Cf. A347113, A347313, A349411.

c[] = 0; j = m = 2; m = 1 + {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, 245}]][[-1, -1]]; Position[m, ?PrimeQ][[All, 1]]

cp's OEIS Frontend

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

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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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A348777 a(n) = term in A347113 that precedes the n-th prime. Set a(n) = -1 if prime(n) does not appear in A347113.

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

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A349412 Positions of prime j = A347113(i)+1.

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