%I #26 Nov 20 2023 00:03:10

%S 16,24,24,45,48,49,120,120,125,189,240,240,350,350,350,350,374,494,

%T 494,714,714,714,714,825,832,1078,1078,1078,1078,1425,1440,1440,1856,

%U 2175,2175,2175,2175,2175,2175,2175,2870,2870,2870,2871,2880,2880,2880,3219

%N a(n) is the largest integer m such that the product of n consecutive integers starting at m is divisible by at most n primes.

%C Each term is only conjectured and has been verified up to 10^6.

%C Note a(2) is undefined if there are infinitely many Mersenne primes.

%e For example, a(3) = 16 because 16 * 17 * 18 = 2^5 * 3^2 * 17 admits only three prime divisors (2, 3, and 17) and appears to be the largest product of three consecutive integers with the property.

%o (SageMath)

%o for r in range(3, 100):

%o history = []

%o M = 0

%o for n in range(1, 100000):

%o primes = {p for p, _ in factor(n)}

%o history.append(primes)

%o history = history[-r:]

%o total = set()

%o for s in history: total |= s

%o # Skip if too many primes.

%o if len(total) > r: continue

%o if n > M: M = n

%o print(r, M-r+1)

%Y Cf. A002182, A045619, A163264, A164799, A239035, A244656, A006549.

%K nonn

%O 3,1

%A _Onno M. Cain_, Sep 06 2019