OFFSET
1,2
COMMENTS
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.
LINKS
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
EXAMPLE
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.
MAPLE
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
MATHEMATICA
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 *)
PROG
(PARI) 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
(Python)
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Grant Olson, Aug 18 2021
EXTENSIONS
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
STATUS
approved