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

[Eric Angelini]

[My last post of the day, Olivier (Gérard), I know what you're about to say !-]

The _Question_ at the bottom of my last msg (visible herunder), 
could be asked for an equivalent seq dealing with primes:

> What is the minimal value of the constant k such that there 
> exists a sequence P where there are a(n) primes < k*a(n)?
> (k not necessarily being an integer)

I've tested unsuccessfully k = 2, 3 and 4 -- they all fail at some point
when using the greedy algorithm.

Example with k=4 [fail for a(9) = 13]:

n = 1 2 3 4 5 6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
P = 1,2,4,5,6,8,11,12,13,14,15,16,17,18,20,21,22,23,24,25,26,27,28,29,30,31,
prime *   *      *     *     *     *              *                 *     *

(sequitur)
n = 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49
P = 32,33,34,35,36,37,38,39,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55...
primes              *        *     *           *                 *

a(1) = 1  and indeed there  is 1 prime   <  4 in P [2];
a(2) = 2  and indeed there are 2 primes  <  8 in P [2,5];
a(3) = 4  and indeed there are 4 primes  < 16 in P [2,5,11,13];
a(4) = 5  and indeed there are 5 primes  < 20 in P [2,5,11,13,17];
a(5) = 6  and indeed there are 6 primes  < 24 in P [2,5,11,13,17,23];
a(6) = 8  and indeed there are 8 primes  < 32 in P [2,5,11,13,17,23,29,31];
a(7) = 11 and indeed there are 11 primes < 44 in P [2,5,11,13,17,23,29,31,37,41,43];
a(8) = 12 and indeed there are 12 primes < 48 in P [2,5,11,13,17,23,29,31,37,41,43,47];
a(9) = 13 and ... NO ... there aren't 13 primes < 48 in P.

Best,
É.









-----Message d'origine-----
De : SeqFan [mailto:seqfan-bounces at list.seqfan.eu] De la part de Eric Angelini
Envoyé : lundi 13 juillet 2015 17:14
À : Sequence Fanatics Discussion list
Objet : [LIKELY_SPAM][seqfan] Non-primes quantities -- and a question

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.


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