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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). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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.
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.
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 *)
PROG
(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
CROSSREFS
Sequence in context: A257347 A143345 A111679 * A087174 A071963 A224075
KEYWORD
nonn
AUTHOR
Scott R. Shannon, Dec 03 2021
STATUS
approved

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Last modified March 28 08:02 EDT 2024. Contains 371236 sequences. (Running on oeis4.)