[seqfan] Non-primes quantities -- and a question

[Eric Angelini]

Hello SeqFans,
U is the lexicographically first sequence such that 
there are a(n) non prime terms < 3*a(n) in U:

n =  1 2 3 4 5 6 7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
U =  1,2,4,5,6,7,8,11,12,13,15,17,18,19,21,23,24,25,26,33,36,37,39,...
np   *   *   *   *     *     *     *     *     *  *  *  *  *     *

Explanation:
a(1) = 1 and indeed there  is 1 non-prime  < 3 in U [1];
a(2) = 2 and indeed there are 2 non-primes < 6 in U [1 and 4];
a(3) = 4 and indeed there are 4 non-primes < 12 in U [1,4 6 and 8];
a(4) = 5 and indeed there are 5 non-primes < 15 in U [1,4,6,8 and 12];
a(5) = 6 and indeed there are 6 non-primes < 18 in U [1,4,6,8,12 and 15];
a(6) = 7 and indeed there are 7 non-primes < 21 in U [1,4,6,8,12,15 and 18];
a(7) = 8 and indeed there are 8 non-primes < 24 in U [1,4,6,8,12,15,18 and 21];
a(8) = 11 and indeed there are 11 non-primes < 33 in U [1,4,6,8,12,15,18,21,24,25 and 26];
a(9) = 12 etc.

Question:
What is the minimal k such that there exist a sequence V 
where there are a(n) non prime terms < k*a(n) in V.

Of course we accept that k is not an integer (like k = 2,912500687...)

Best,
É.