A349872 a(1) = 2; for n > 1, a(n) is the smallest unused number > 1 such that none of the previous a(n) terms divide a(n).

2, 3, 5, 7, 11, 4, 13, 17, 6, 19, 23, 9, 29, 10, 8, 31, 37, 41, 14, 15, 43, 12, 47, 53, 59, 21, 61, 22, 25, 67, 18, 16, 71, 26, 20, 73, 79, 83, 33, 27, 35, 89, 34, 97, 101, 103, 24, 28, 38, 39, 30, 107, 109, 49, 113, 127, 131, 46, 137, 51, 55, 139, 149, 151, 32, 45, 157, 36, 42, 57, 163, 58, 44

Offset: 1

Author

Scott R. Shannon, Dec 03 2021

Keywords

• nonn

Comments

Slightly under two-thirds of the terms are between the lines a(n) = n/2 and a(n) = n; in the range studied all of these numbers are composite. The remaining terms, all of which are primes, lie approximately on a curve that starts with a slope near 2 that slowly increases. See the linked image.

There are no fixed points up to 10000 terms so it is likely none exist.

For many even terms a(n) where n > a(n) it is found that a(n - a(n) - 1) = a(n)/2. The first even term where that is not the case is a(113) = 68, as a(113 - 68 - 1) = a(44) = 97, not 34. In this case a(43) = 34.

For a given number k the longest possible sequence of unique numbers that contains a number every k terms that divides k is finite, ~ 2*sqrt(k)*k; this implies all numbers > 1 eventually appear.

Links

• Scott R. Shannon, Image of the first 10000 terms. The green line is a(n) = n.

Crossrefs

Cf. A027750, A093714, A064413, A000040.

Programs

Mathematica

a[1]=2;a[n_]:=a[n]=(k=2;While[MemberQ[s=Array[a,n-1],k]||Or@@(IntegerQ/@(k/s[[-If[k>=n,n-1,k];;]])),k++];k);Array[a,73] (* Giorgos Kalogeropoulos, Dec 03 2021 *)

Python

def aupton(terms):
    alst, aset = [2], {2}
    for n in range(2, terms+1):
        k = 2
        while k in aset or any(k%j == 0 for j in alst[-k:]): k += 1
        alst.append(k); aset.add(k)
    return alst
print(aupton(73)) # Michael S. Branicky, Dec 03 2021

Formula

a(2) = 3 as the previous term 2 does not divide 3.

a(6) = 4 as none of the previous four terms, 3, 5, 7, 11, divide 4.

a(9) = 6 as none of the previous six terms, 5, 7, 11, 4, 13, 17, divide 6.