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

[israel at math.ubc.ca]

Since pi(n)/n -> 0 as n -> infinity, all such sequences are doomed to be 
finite.

Cheers,
Robert Israel

On Jul 13 2015, Eric Angelini wrote:

> [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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