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

1,2

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.

Links

Table of n, a(n) for n=1..72.

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.

Example

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

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

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 *)

Prog

(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

Crossrefs

Cf. A085229, A103683, A350359.

Sequence in context: A338642 A263792 A263411 * A369800 A367291 A367289

Adjacent sequences: A352947 A352948 A352949 * A352951 A352952 A352953

Keyword

nonn

Author

David James Sycamore, Apr 10 2022

Status

approved