A352950 Lexicographically earliest infinite sequence of distinct nonnegative integers commencing 1,3,5,7 such that any four consecutive terms are pairwise coprime.

1, 3, 5, 7, 2, 9, 11, 13, 4, 15, 17, 19, 8, 21, 23, 25, 16, 27, 29, 31, 10, 33, 37, 41, 14, 39, 43, 47, 20, 49, 51, 53, 22, 35, 57, 59, 26, 55, 61, 63, 32, 65, 67, 69, 28, 71, 73, 45, 34, 77, 79, 75, 38, 83, 89, 81, 40, 91, 97, 87, 44, 85, 101, 93, 46, 95, 103, 99, 52, 107, 109, 105

Offset: 1

Author

David James Sycamore, Apr 10 2022

Keywords

• nonn

Comments

The pairwise coprime relations found in the first four odd numbers 1,3,5,7 are preserved throughout in any run of four consecutive terms.

a(4n+5) is always even (and < a(4n+2)); n>=0.

The plot exhibits two distinct rays at first (upper/odd, lower/even), with no terms divisible by 6 until a(229), at which point the even ray switches to producing just 28 multiples of 6 until a(337)=168. At this point the original even ray is re-established, the odd ray divides into two (quasi-parallel) rays, and no further multiples of 6 are seen. Therefore it seems very unlikely that the sequence is a permutation of the nonnegative integers.

Primes p other than p = 2 appear in their natural order.

Examples

			3,5,7 are pairwise coprime and 2 is the smallest unused number coprime to all of them, therefore a(5)=2.
		

Links

• Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^14, highlighting primes in green, numbers divisible by 6 in gold, other even numbers in red, odd numbers divisible by 3 in blue.

Crossrefs

Cf. A085229, A103683, A350359.

Programs

Maple

ina := proc(n) false end: # adapted from code for A103683
a := proc (n) option remember; local k;
if n < 5 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 or igcd(k, a(n-3)) <> 1
do
end do
end if; ina(k):= true; k
end proc;
seq(a(n), n = 1 .. 100);

Mathematica

Block[{a, c, k, m, u, nn}, nn = 86; c[] = 0; MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, First[#2]}] &, {1, 3, 5, 7}]; u = 2; Do[k = u; m = LCM @@ Array[a[i - #] &, 3]; 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, 5, nn}]; Array[a, nn] ] (* _Michael De Vlieger, Apr 12 2022 *)

Python

from math import gcd
from itertools import islice
def agen(): # generator of terms
    aset, b, c, d = {1, 3, 5, 7}, 3, 5, 7
    yield from [1, b, c, d]
    while True:
        k = 1
        while k in aset or any(gcd(t, k) != 1 for t in [b, c, d]): k+= 1
        b, c, d = c, d, k
        aset.add(k)
        yield k
print(list(islice(agen(), 107))) # Michael S. Branicky, Apr 10 2022