A309366 -id:A309366 - cp's OEIS frontend

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.

3, 7, 433, 9257821

Offset: 1

Peter Munn, Feb 01 2017

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.

			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.

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

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

A126283 Largest number k for which the n-th prime is the median of the largest prime dividing the first k integers.

Original entry on oeis.org

4, 18, 40, 76, 116, 182, 246, 330, 426, 532, 652, 770, 904, 1058, 1210, 1386, 1560, 1752, 1956, 2162, 2394, 2640, 2894, 3150, 3422, 3680, 3984, 4302, 4628, 4974, 5294, 5650, 5914, 6006, 6372, 6746, 7146, 7536, 7938, 8386, 8794, 9222, 9702, 10156

Offset: 1

Mark Thornquist (mthornqu(AT)fhcrc.org) & Robert G. Wilson v, Dec 15 2006

nonn

a(14) = 1058 is the first term where a(n) exceeds A290154(n). - Peter Munn, Aug 02 2019

			a(1)=4 because the median of {2,3,2} = {2, *2*,3} is 2 (the * surrounds the median) and for any number greater than 4 the median is greater than 2.
a(1)=18 because the median of {2,3,2,5,3,7,2,3,5,11,3,13,7,5,2,17,3} = {2,2,2,2,3,3,3,3, *3*,5,5,5,7,7,11,13,17}.

Other sequences about medians of prime factors: A124202, A126282, A281889, A284411, A290154, A308904.

Cf. A000040, A309366.

t = Table[0, {100}]; lst = {}; Do[lpf = FactorInteger[n][[ -1, 1]]; AppendTo[lst, lpf]; mdn = Median@lst; If[PrimeQ@ mdn, t[[PrimePi@mdn]] = n], {n, 2, 10^4}]; t

cp's OEIS Frontend

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.

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