A352849 a(n) is the least k not already in the sequence such that k is pairwise coprime to a(n-1) and a(n-2), starting with a(1) = 1, a(2) = 3, and a(3) = 5.

1, 3, 5, 2, 7, 9, 4, 11, 13, 6, 17, 19, 8, 15, 23, 14, 25, 27, 16, 29, 21, 10, 31, 33, 20, 37, 39, 22, 35, 41, 12, 43, 47, 18, 49, 53, 24, 55, 59, 26, 45, 61, 28, 51, 65, 32, 57, 67, 34, 63, 71, 38, 69, 73, 40, 77, 79, 30, 83, 89, 36, 85, 91, 44, 75, 97, 46, 81

Offset: 1

Author

Michael De Vlieger, Apr 14 2022

Keywords

• nonn

Comments

Sequence begins with 1 and the first 2 odd primes.

a(3k+1) is even for k > 0 as consequence of definition and since 2 is the smallest prime and numbers are either even or odd. Unlike A085229, even numbers in this sequence do not appear in order. Hence a(3k) and a(3k+2) are odd.

The smallest missing number u is even, and there is a smallest missing odd number v that applies to a(3k) and a(3k+2).

Let q be odd and prime. In a given interval i <= n <= j, we either have 2q | a(n) or we have q | a(n) odd.

Regarding 6 | a(n), there are phases i <= n <= j where 6 | a(3k+1) and no a(n) mod 6 = 3 appear. These begin when 3 | a(3k+1) and prevent the entry of 3 | v. Whereupon all u such that 6 | u have been consumed, 3 | a(3k+r), r != 1 occurs, and we move into a phase where we have a(n) mod 6 = 3, but no 6 | a(3k+1) appear. This occurs until all v such that 3 | v have been consumed, and 3 | a(3k+1) once again.

Conjecture: the sequence is a permutation of the natural numbers.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Annotated log-log scatterplot of a(n), n = 1..2^12, showing odd numbers congruent to 1 or 5 (mod 6) in green, odd numbers congruent to 3 (mod 6) in blue, even numbers congruent to 2 or 4 (mod 6) in red, and numbers divisible by 6 in amber.

Crossrefs

Cf. A085229, A352950.

Programs

Maple

ina := proc(n) false end: # adapted from code for A352950
a := proc (n) option remember; local k;
if n < 4 then k := 2*n-1
else for k from 2 while ina(k) or igcd(k, a(n-1)) <> 1 or igcd(k, a(n-2)) <>1
do
end do
end if; ina(k):= true; k
end proc:
seq(a(n), n = 1 .. 100); # -David James Sycamore, Apr 17 2022

Mathematica

nn = 66, c[_] = 0; Array[Set[{a[#1], c[#2]}, {#2, #1}] & @@ {#, 2 # - 1} &, 3]; u = 2; Do[k = u; m = LCM @@ Array[a[i - #] &, 2]; While[Nand[c[k] == 0, CoprimeQ[m, k]], k++]; Set[{a[i], c[k]}, {k, i}]; If[a[i] == u, While[c[u] > 0, u++]], {i, 4, nn}]; Array[a, nn]

Python

from math import gcd
from itertools import islice
def agen(): # generator of terms
    aset, b, c = {1, 3, 5}, 3, 5
    yield from [1, b, c]
    while True:
        k = 1
        while k in aset or any(gcd(t, k) != 1 for t in [b, c]): k+= 1
        b, c = c, k
        aset.add(k)
        yield k
print(list(islice(agen(), 68))) # Michael S. Branicky, Apr 14 2022