A355212 A variant of the EKG sequence (A064413) where the least value not yet in the sequence appears as soon as possible.

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, 19, 57, 21, 24, 22, 46, 23, 115, 25, 30, 26, 36, 27, 42, 28, 58, 29, 899, 31, 62, 32, 74, 37, 148, 38, 40, 82, 41, 1763, 43, 86, 44, 48, 45, 141, 47, 329, 49, 56, 50

Offset: 1

Author

Rémy Sigrist, Jun 24 2022

Keywords

• nonn

Comments

To build the sequence:

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

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

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

then a(n+1) = v,

- otherwise:

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

and gcd(w, v) > 1,

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

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

The construction is similar to that of A352713.

Examples

			The first terms are (stars correspond to "w" terms):
  n   a(n)  w
  --  ----  -
   1     1
   2     2
   3     6  *
   4     3
   5    12  *
   6     4
   7    10  *
   8     5
   9    35  *
  10     7
  11    14  *
  12     8
  13    18  *
  14     9
  15    33  *
  16    11
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program
• Index entries for sequences related to EKG sequence
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A064413, A352713, A355213 (inverse).

Programs

PARI

\\ See Links section.

Python

from math import gcd
from itertools import count, islice
def agen(): # generator of terms
    aset, an, v = {1, 2}, 2, 3; yield from [1, 2]
    for n in count(3):
        if gcd(an, v) == 1:
            w = v + 1
            while w in aset or gcd(an, w) == 1 or gcd(w, v) == 1: w += 1
            aset.add(w); yield w
        an = v; aset.add(an); yield an
        while v in aset: v += 1
print(list(islice(agen(), 65))) # Michael S. Branicky, Jun 24 2022