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A308905
Number of numbers k such that exactly half the numbers in [1..k] are prime(n)-smooth.
1
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
1,1
COMMENTS
When a(n)=1, A290154(n) = A308904(n). Values of n at which this occurs begin 2, 3, 6, 8, 10, 11, 13, 16, 25, 27, 47, 56, 60, 64, 65, 75, 81, 99, ... Do they tend to occur less frequently as n increases?
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
Sequence in context: A332402 A263284 A332404 * A158471 A158472 A198895
KEYWORD
nonn
AUTHOR
Jon E. Schoenfield, Jun 29 2019
STATUS
approved