

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(n1) + 1.


33



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
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OFFSET

1,2


COMMENTS

Alternative definition: Lexicographically earliest sequence of distinct positive numbers such that a(n) != a(n1)+1 and gcd(a(n1)+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



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(n1)+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:


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[n1]+1); ); va; } \\ Michel Marcus, Aug 21 2021
(Python)
from math import gcd
A347113_list, nset, m = [1], {1}, 2
for _ in range(100):
k = m
while k == j or gcd(k, j) == 1 or k in nset:
k += 1
nset.add(k)
while m in nset:


CROSSREFS



KEYWORD

nonn


AUTHOR



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



