interesting sequence
Jeffrey Shallit
elvis at graceland.uwaterloo.ca
Tue Jan 25 19:25:00 CET 2000
Here's an interesting sequence for which I don't know the next
term:
2, 21, 1310, ?
The sequence (S_n) is defined as follows:
S_n = the least positive integer r such that there exists
an integer s, 0 <= s < r
gcd(r-i, s-j) > 1
for all integers i, j with 0 <= i, j < n.
Then
S_1 = 2 because we can choose r = 2, s = 0
S_2 = 21 because we can choose r = 21, s = 15
S_3 = 1310 because we can choose r = 1310, s = 1276
By brute force search I know that S_4 > 410,000. And also
I know by constructing the pair (r, s) = (477742707, 172379781)
that S_4 <= 477742707.
It's also possible to give asymptotic estimates on the growth
rate of S_n. For example, I can show that
S_n < e^{(1+o(1)) 2 n^2 log n}
The sequence S_n is also related to Jacobsthal's function g(n).
Jeffrey Shallit
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