A339743 a(n) is the least k > 0 such that 1+k, 1+2*k, ..., 1+n*k are pairwise coprime.

1, 1, 2, 4, 6, 6, 6, 30, 30, 60, 60, 210, 210, 210, 210, 210, 210, 210, 2310, 2310, 2310, 2310, 18480, 120120, 120120, 150150, 150150, 150150, 150150, 660660, 1531530, 2492490, 3063060, 3063060, 4594590, 38798760, 38798760, 38798760, 38798760, 38798760, 48498450, 193993800

Offset

1,3

Comments

In other words, a(n) is the least k such that A339749(k) >= n.

From David A. Corneth, Jan 01 2021: (Start)

If 2*p < n then p | a(n) for prime p. Proof: Suppose for such p, we don't have p | a(n). Then for some 1 <= k <= n/2 we have p | 1 + a(n)*k and so we have p | 1 + a(n) * (k + p) as well and gcd(1 + a(n) * k, 1 + a(n) * (k + p)) >= p > 1. Contradiction.

This sequence is weakly increasing. Proof: Let m > n. Then if a(m) < a(n) then by construction a(n) = a(m). Contradiction.

Let f(n) = Product_{i = 1..pi(n/2)} and let c(n) be a candidate for a(n) such that f(n) | c(n) where pi = A000720. If for some 1 + m*c(n) and 1 + (m + t)*c(n) we have gcd(1 + m * c(n), 1 + (m + t)*c) > 1 then n / 2 < t < n and t is prime. (End)

Links

David A. Corneth, Table of n, a(n) for n = 1..200

Patrick Nicodemus, Arithmetic progressions of coprime natural numbers, Mathematics Stack Exchange, Dec 15 2020.

Example

For n = 5:
- A339749(1) = 2 < 5,
- A339749(2) = 3 < 5,
- A339749(3) = 2 < 5,
- A339749(4) = 4 < 5,
- A339749(5) = 2 < 5,
- A339749(6) = 7 >= 5,
- so a(5) = 6.

Prog

(PARI) { n = 1; for (k=1, 38798760, p = 1; for (m=1, oo, if (gcd(p, 1+m*k)>1, break, p *= 1+m*k; if (m==n, print1 (k ", "); n++)))) }

Crossrefs

Cf. A000720, A339749, A339759.

Sequence in context: A078611 A211376 A278249 * A131450 A114218 A133691

Adjacent sequences: A339740 A339741 A339742 * A339744 A339745 A339746

Keyword

nonn

Author

Rémy Sigrist, Dec 15 2020

Status

approved