A281889 - OEIS

%I #50 Feb 06 2020 12:19:01

%S 3,7,433,9257821

%N a(n) is the least integer k such that more than half of all integers are divisible by a product of n integers chosen from 2..k.

%C The n chosen integers need not be distinct.

%C By "more than half of all integers" we mean more precisely "more than half of the integers in -m..m, for all sufficiently large m (depending on n)", and similarly with 1..m for "more than half of all positive integers".

%C Equivalently, a(n) is the least prime p such that more than half of all positive integers can be written as a product of primes of which n or more are not greater than p. (In this sense, a(n) might be called the median n-th least prime factor of the integers.)

%C The number of integers that satisfy the "product of primes" criterion for p = prime(m) is the same in every interval of primorial(m)^n integers and is A281891(m,n). Primorial(m) = A002110(m), product of the first m primes.

%C a(n) is the least k = prime(m) such that 2 * A281891(m,n) > A002110(m)^n.

%C a(n) is the least k such that more than half of all positive integers equate to the volume of an orthotope with integral sides at least n of which are orthogonal with length between 2 and k inclusive.

%C The next term is estimated to be a(5) ~ 3*10^18.

%e For n=1, we have a(1) = 3 since for all m > 1, more than half of the integers in -m..m are divisible by an integer chosen from 2..3, i.e., either 2 or 3. We must have a(1) > 2, because the only integer in 2..2 is 2, but in each interval -2m-1..2m+1, only 2m+1 integers are even, so 2 is not a divisor of more than half of all integers in the precise sense given above.

%Y Other sequences about medians of prime factors: A126282, A126283, A284411, A290154.

%Y Cf. A002110, A014673, A016088, A027746, A115561, A281891, A309366.

%K nonn,hard,more

%O 1,1

%A _Peter Munn_, Feb 01 2017