A308905 Number of numbers k such that exactly half the numbers in [1..k] are prime(n)-smooth.

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

Author

Jon E. Schoenfield, Jun 29 2019

Keywords

• nonn

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?

Examples

			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

Cf. A290154, A308904.