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%I #8 Jun 29 2019 23:04:43
%S 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,
%T 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,
%U 2,10,5,4,6,9,7,1,3,8,8,6,6,1,3,4,2,2,2
%N Number of numbers k such that exactly half the numbers in [1..k] are prime(n)-smooth.
%C 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?
%e 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:
%e .
%e Number m
%e 2-smooth of 2-smooth
%e numbers numbers
%e k in [1..k] in [1..k] m/k
%e == ============ =========== ===============
%e 1 {1} 1 1/1 = 1.000000
%e 2 {1, 2} 2 2/2 = 1.000000
%e 3 {1, 2} 2 2/3 = 0.666667
%e 4 {1, 2, 4} 3 3/4 = 0.750000
%e 5 {1, 2, 4} 3 3/5 = 0.600000
%e 6 {1, 2, 4} 3 3/6 = 0.500000 = 1/2
%e 7 {1, 2, 4} 3 3/7 = 0.428571
%e 8 {1, 2, 4, 8} 4 4/8 = 0.500000 = 1/2
%e 9 {1, 2, 4, 8} 4 4/9 = 0.444444
%e 10 {1, 2, 4, 8} 4 4/10 = 0.400000
%e .
%e 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.
%e 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.
%Y Cf. A290154, A308904.
%K nonn
%O 1,1
%A _Jon E. Schoenfield_, Jun 29 2019