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.
Original entry on oeis.org
3, 7, 433, 9257821
Offset: 1
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.
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
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}.
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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
A308904
Largest number k such that exactly half the numbers in [1..k] are prime(n)-smooth.
Original entry on oeis.org
8, 20, 42, 84, 128, 184, 256, 332, 432, 534, 654, 784, 906, 1060, 1226, 1388, 1568, 1772, 1962, 2166, 2420, 2646, 2928, 3162, 3424, 3692, 3986, 4308, 4630, 4984, 5296, 5658, 6008, 6376, 6750, 7156, 7540, 7958, 8388, 8806, 9226, 9704, 10170, 10634, 11140, 11664
Offset: 1
The 2-smooth numbers are 1, 2, 4, 8, 16, 32, ... (A000079, the powers of 2), so exactly half of the 8 numbers in the interval [1..8] are 2-smooth numbers: the 8/2 = 4 numbers 1, 2, 4, and 8. For all numbers k > 8, the number of 2-smooth numbers in [1..k] is less than k/2, so 8 is the largest k at which the number of 2-smooth numbers in [1..k] is exactly k/2, so a(1)=8. (The smallest k at which the number of 2-smooth numbers in [1..k] is exactly k/2 is A290154(1) = 6.)
The 3-smooth numbers are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, ... (A003586). It can be shown that k=20 is the only number k such that exactly half of the numbers in the interval [1..k] are 3-smooth. Since k=20 is the only such number, 20 is both a(2) and A290154(2).
A308905
Number of numbers k such that exactly half the numbers in [1..k] are prime(n)-smooth.
Original entry on oeis.org
2, 1, 1, 4, 5, 1, 4, 1, 3, 1, 1, 2, 1, 2, 7, 1, 4, 4, 3, 2, 5, 3, 6, 6, 1, 4, 1, 3, 2, 5, 3, 3, 2, 2, 2, 5, 4, 7, 8, 7, 2, 6, 5, 3, 13, 10, 1, 9, 2, 6, 3, 2, 8, 4, 4, 1, 11, 3, 3, 1, 7, 2, 4, 1, 1, 5, 4, 2, 10, 5, 4, 6, 9, 7, 1, 3, 8, 8, 6, 6, 1, 3, 4, 2, 2, 2
Offset: 1
For n=1: prime(1)=2, and the 2-smooth numbers are 1, 2, 4, 8, 16, 32, ... (A000079, the powers of 2), so for k = 1..10, the number of 2-smooth numbers in the interval [1..k] increases as follows:
.
Number m
2-smooth of 2-smooth
numbers numbers
k in [1..k] in [1..k] m/k
== ============ =========== ===============
1 {1} 1 1/1 = 1.000000
2 {1, 2} 2 2/2 = 1.000000
3 {1, 2} 2 2/3 = 0.666667
4 {1, 2, 4} 3 3/4 = 0.750000
5 {1, 2, 4} 3 3/5 = 0.600000
6 {1, 2, 4} 3 3/6 = 0.500000 = 1/2
7 {1, 2, 4} 3 3/7 = 0.428571
8 {1, 2, 4, 8} 4 4/8 = 0.500000 = 1/2
9 {1, 2, 4, 8} 4 4/9 = 0.444444
10 {1, 2, 4, 8} 4 4/10 = 0.400000
.
It is easy to show that, for all k > 8, fewer than half of the numbers in [1..k] are 2-smooth, so there are only 2 values of k, namely, k=6 and k=8, at which exactly half of the numbers in the interval [1..k] are 2-smooth numbers, so a(1)=2.
For n=2: prime(2)=3, and the 3-smooth numbers are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, ... (A003586). It can be shown that k=20 is the only number k such that exactly half of the numbers in the interval [1..k] are 3-smooth. Since there is only 1 such number k, a(2)=1.
Showing 1-4 of 4 results.
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