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A341679
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a(1) = 1; for n > 1, a(n) = n divided by the most recently appearing divisor of n in all previous terms.
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4
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1, 2, 3, 2, 5, 3, 7, 4, 3, 2, 11, 6, 13, 7, 5, 8, 17, 3, 19, 4, 7, 2, 23, 12, 5, 13, 9, 14, 29, 6, 31, 16, 11, 17, 7, 6, 37, 19, 3, 8, 41, 14, 43, 4, 15, 2, 47, 24, 7, 25, 17, 26, 53, 27, 5, 8, 19, 29, 59, 12, 61, 31, 9, 8, 13, 33, 67, 4, 23, 14, 71, 18, 73, 37, 15, 19, 11, 6, 79, 20, 9, 41, 83
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OFFSET
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1,2
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COMMENTS
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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)
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LINKS
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FORMULA
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a(n) = n if n is prime.
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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