OFFSET
1,1
COMMENTS
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn
AUTHOR
Jon E. Schoenfield, Jun 29 2019
STATUS
approved