A355212 - OEIS

%I #14 Sep 03 2024 15:03:45

%S 1,2,6,3,12,4,10,5,35,7,14,8,18,9,33,11,143,13,39,15,20,16,34,17,323,

%T 19,57,21,24,22,46,23,115,25,30,26,36,27,42,28,58,29,899,31,62,32,74,

%U 37,148,38,40,82,41,1763,43,86,44,48,45,141,47,329,49,56,50

%N A variant of the EKG sequence (A064413) where the least value not yet in the sequence appears as soon as possible.

%C To build the sequence:

%C - we start with a(1) = 1 and a(2) = 2, and then repeatedly:

%C - let a(n) be the last known term and v the least value not yet in the sequence,

%C - if gcd(a(n), v) > 1

%C   then a(n+1) = v,

%C - otherwise:

%C   - let w be the least value not yet in the sequence such that gcd(a(n), w) > 1

%C     and gcd(w, v) > 1,

%C   - then a(n+1) = w and a(n+2) = v.

%C This sequence is a permutation of the positive integers with inverse A355213.

%C The construction is similar to that of A352713.

%H Rémy Sigrist, <a href="/A355212/b355212.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A355212/a355212.gp.txt">PARI program</a>

%H <a href="/index/Ed#EKG">Index entries for sequences related to EKG sequence</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%e The first terms are (stars correspond to "w" terms):

%e   n   a(n)  w

%e   --  ----  -

%e    1     1

%e    2     2

%e    3     6  *

%e    4     3

%e    5    12  *

%e    6     4

%e    7    10  *

%e    8     5

%e    9    35  *

%e   10     7

%e   11    14  *

%e   12     8

%e   13    18  *

%e   14     9

%e   15    33  *

%e   16    11

%o (PARI) \\ See Links section.

%o (Python)

%o from math import gcd

%o from itertools import count, islice

%o def agen(): # generator of terms

%o     aset, an, v = {1, 2}, 2, 3; yield from [1, 2]

%o     for n in count(3):

%o         if gcd(an, v) == 1:

%o             w = v + 1

%o             while w in aset or gcd(an, w) == 1 or gcd(w, v) == 1: w += 1

%o             aset.add(w); yield w

%o         an = v; aset.add(an); yield an

%o         while v in aset: v += 1

%o print(list(islice(agen(), 65))) # _Michael S. Branicky_, Jun 24 2022

%Y Cf. A064413, A352713, A355213 (inverse).

%K nonn

%O 1,2

%A _Rémy Sigrist_, Jun 24 2022