除第一项外,所有项均等于模15的7、11、13或14。如果我们定义
Pi(N,b)=#{p素数,p<=N,p==b(mod 15)};
Q(N)=#{p素数,2<p<=N,p在这个序列}中,
然后根据Artin猜想,Q(N)~(94/95)*C*N/log(N)~(188/95)*C*(Pi(N,7)+Pi=A005596号是阿廷常数。
推测:如果我们进一步定义
Q(N,b)=#{p素数,p<=N,p==b(mod 15),p在这个序列中},
那么我们有:
Q(N,7)~(10/47)*Q(N)~(80/95)*C*Pi(N,七);
Q(N,11)~(12/47)*Q(N)~(96/95)*C*Pi(N,十一);
Q(N,13)~(10/47)*Q(N)~(80/95)*C*Pi(N,十三);
Q(N,14)~(15/47)*Q(N)~(120/95)*C*Pi(N,十四)。
数值验证上限tp N=10^8:
|Q(N,7)|Q(N,11)|Q
-------------+---------+---------+---------+---------+---------
N=10^3|14|18|13|19|64
Q(N,*)/Q(N)|0.21875|0.28125|0.20313|0.29688|1.0000
-------------+---------+---------+---------+---------+---------
N=10^4|97|115|90|138|440
Q(N,*)/Q(N)|0.22045|0.26136|0.20455|0.31364|1.0000
-------------+---------+---------+---------+---------+---------
N=10^5|753|891|750|1129|3523
Q(N,*)/Q(N)|0.21374|0.25291|0.21289|0.32047|1.0000
-------------+---------+---------+---------+---------+---------
N=10^6|6153|7395|6176|9247|28971
Q(N,*)/Q(N)|0.21238|0.25526|0.21318|0.31918|1.0000
-------------+---------+---------+---------+---------+---------
N=10^7|52427|62973|52368|78398|246166
Q(N,*)/Q(N)|0.21297|0.25582|0.21273|0.31848|1.0000
-------------+---------+---------+---------+---------+---------
N=10^8|453936|544551|453699|680226|2132412
Q(N,*)/Q(N)|0.21287|0.25537|0.21276|0.31899|1.0000
-------------+---------+---------+---------+---------+---------
推测|0.21277|0.25532|0.21277 |0.31915|1.00000
(结束)