OFFSET
1,2
COMMENTS
From Michael De Vlieger, Mar 03 2021: (Start)
Records are 1 and the primes, i.e., A008578.
1 is the minimum, with occasions of the term 2 local minima.
a(p)=p since a(1)=1, the empty product, is the only available divisor in a(n) for 1 <= n <= p for any prime p, since p itself does not yet appear in the sequence, and p is coprime to all smaller n, i.e., gcd(p, n) = 1 for all n < p. As a consequence of a(p) = p, a(1) = 1 is the only appearance of 1 in the sequence.
a(p^2) = p, and generally, a(p^k) = p^e, 1 <= e < k, 1 < a(c) < c for composite c.
Trajectories visible in the scatterplot of a(n) pertain to d = n/a(n) and have origin (d, 1) and slope 1/d. Noncomposite trajectories d appear more continuous than composite trajectories, which exhibit a quasi-regular, exponential pattern of interruptions.
The plot of n at (x,y) = (a(n), n/a(n)) "unfolds" the scatterplot of this sequence. (End)
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated plot of n at (x,y) = (a(n), n/a(n)) for 1 <= x <= 139 and 1 <= y <= 99, showing noncomposite d = n/a(n) in red and composite d in blue, with n such that both a(n) and d noncomposite in black.
Michael De Vlieger, Plot of n at (x,y) = (a(n), n/a(n)) for 1 <= x <= 960 and 1 <= y <= 960.
Michael De Vlieger, Log-log plot of a(n) for 1 <= n <= 2^20.
Scott R. Shannon, Image for n=1..1000000.
FORMULA
a(n) = n if n is prime.
EXAMPLE
a(4) = 2 as a(2) = 2 is the most recently occurring divisor of 4, thus a(4) = 4/2 = 2.
a(5) = 5 as the only divisor of 5 in the sequence is 1, thus a(5) = 5/1 = 5.
a(10) = 2 as a(5) = 5 is the most recently occurring divisor of 10, thus a(10) = 10/5 = 2.
MATHEMATICA
Block[{a = {1}, k}, Do[k = 1; While[Mod[i, a[[-k]]] != 0, k++]; AppendTo[a, i/a[[-k]] ], {i, 2, 83}]; a] (* Michael De Vlieger, Feb 17 2021 *)
(* Second, faster program with memoized last indices of d | n *)
Block[{a = {1}, c, k}, c[1] = 1; Monitor[Do[AppendTo[a, Set[k, i/MaximalBy[Map[If[! IntegerQ@ c[#], {#, 0}, {#, c[#]}] &, Divisors[i]], Last][[1, 1]] ]]; c[k] = i , {i, 2, 10^4}], i]; a] (* Michael De Vlieger, Mar 03 2021 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Scott R. Shannon, Feb 17 2021
STATUS
approved