A308904 Largest number k such that exactly half the numbers in [1..k] are prime(n)-smooth.

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

Author

Jon E. Schoenfield, Jun 29 2019

Keywords

• nonn

Comments

Cf. A290154 (Smallest number k such that exactly half the numbers in [1..k] are prime(n)-smooth).

It appears that for most values of n, there exists more than one number k such that exactly half the numbers in [1..k] are prime(n)-smooth; see A308905.

Examples

			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).
		

Crossrefs

Cf. A290154, A308905.