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 A281889 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. 9
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 OFFSET 1,1 COMMENTS The n chosen integers need not be distinct. 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". 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.) 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. a(n) is the least k = prime(m) such that 2 * A281891(m,n) > A002110(m)^n. 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. The next term is estimated to be a(5) ~ 3*10^18. LINKS EXAMPLE 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. CROSSREFS Other sequences about medians of prime factors: A126282, A126283, A284411, A290154. Cf. A002110, A014673, A016088, A027746, A115561, A281891, A309366. Sequence in context: A103317 A104051 A128004 * A086559 A013426 A013422 Adjacent sequences:  A281886 A281887 A281888 * A281890 A281891 A281892 KEYWORD nonn,hard,more AUTHOR Peter Munn, Feb 01 2017 STATUS approved

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Last modified May 28 13:05 EDT 2022. Contains 354115 sequences. (Running on oeis4.)