Hofstadteriana with primes

[Jacques Tramu]

From: "Eric Angelini" <Eric.Angelini at kntv.be>


> 
> Hello SeqFans,
> 
> S = 1   3   8   15  26  39  56  75  98  127  158  195  236  279  326...
> d =   2   5   7   11  13  17  19  23  29   31   37   41   43   47 ... 
> 
> - start S with 1
> - add the smallest prime not yet added and not already present in S
> 
> Question:
> - What could be the ratio primes/composites of S?

Pleese find herewith :

n, S[n] , np = # of primes in S[1..n],  ratio = np/n (%)

1000    3724118 79      7.9000000000
2000    16391590        137     6.8500000000
3000    38857028        194     6.4666666667
4000    71609700        260     6.5000000000
5000    114938340       309     6.1800000000
6000    169071260       354     5.9000000000
7000    234234190       393     5.6142857143
8000    310623308       444     5.5500000000
9000    398353906       500     5.5555555556
10000   497496444       551     5.5100000000
11000   608310496       595     5.4090909091
12000   730829204       642     5.3500000000
13000   865068282       693     5.3307692308
14000   1011175982      743     5.3071428571
15000   1169195340      777     5.1800000000
16000   1339435452      826     5.1625000000
17000   1521736078      863     5.0764705882
18000   1716121198      909     5.0500000000
19000   1922775792      967     5.0894736842
20000   2141711486      1009    5.0450000000
21000   2372927660      1054    5.0190476190
22000   2616641270      1091    4.9590909091
23000   2872744162      1156    5.0260869565
24000   3141343994      1193    4.9708333333

I have a proof that the limit of the ratio is 5%, unfortunately,
the left margin of this sheet of paper is too small ....

regards,
JT