Prime related

[kohmoto]

    These are prime related sequences.
    I think the first one is interesting.

    I searched A000007.
    It exists on OEIS.


    %S A000001 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 
0, 0, 0, 0

    %N A000001     a(n)=Prime(n)+Prime(n+k) , mod 4
                         k=1/2*(Prime(n+1)-Prime(n))

    %C A000001    If {Prime(n), Prime(n+1)} are twin primes, then a(n)=0, 
because k=1, so a(n)=2*Prime(n)+2=0 , mod 4 .
                         For n=31, 61, 73, a(n)=2


                         I think the number of 2 is too few. I supposed that 
the  ratio (#{m | a(m)=2 , m<=n})/(Pi(n)-Pi_2(n)) is almost 1/2, but for 
n=100, ratio=3/(24-8)=3/16.
                         Does any other reason exist?

                         I have no idea to calculate the exact ratio that 
limit k->infinity {N(k)/(Pi(k)-Pi_2(k))} .
                         Where N(k) means number of terms of a(n)=2 up to k 
. N(k)=#{m | a(m)=2 , m<=k} .

    %0 A000001     2,10 ..... sequence starts from n=2



    %S A000002 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 
2, 0, 2, 2
    %N A000002     a(n)=Prime(n)+Prime(n+1) , mod 4
    %C A000002     ratio=9/16 ..... understandable
    %O A000002    2, 3


    %S A000003 0, 2, 2, 0, 2, 0, 0, 0, 2, 2, 2, 0, 0, 2, 0, 0, 2, 2, 2, 2, 
2, 0, 0
    %N A000003     a(n)=Prime(n-1)+Prime(n) , mod 4
                                k=1/2*(Prime(n+1)-Prime(n))
    %O A000003    5,2


    %S A000004 0, 0, 1, 1, 2, 1, 2, 2, 3, 2, 3, 4
    %N A000004     Number of ways of sum such that Prime(n)-1 
=Prime(i)+Prime(j).
    %e A000004              11-1=3+7=5+5 , so a(5)=2 .
    %O A000004    1,5


    %S A000005     0, 0, 0, 1, 2, 2, 4, 2, 3, 4
    %N A000005     Number of ways of sum such that Prime(n) = 
2*Prime(i)+Prime(j).
    %e A000005              11=2*2+7=2*3+5 , so a(5)=2



    %S A000006 2, 3, 11, 29, 226
    %N A000006  a(n)=Sum( Product p_i , {Sum p_i=Prime(n)} , p_i is prime )
    %e A000006   a(5)=2*2*2*2*3+2*2*2*5+2*2*7+2*3*3*3+3*3*5+11


    %S A000007 1, 1, 2, 3, 6, 9, 17
    %N A000007      Number of partitions of a prime into primes.
    %e A000007   a(5)=2+2+2+2+3=2+2+2+5=2+2+7=2+3+3+3=3+3+5=11


    %S A000008 4, 4, 4, 4, 4, 6, 6, 4, 6, 12, 8, 4
    %N A000008    a(n)=(Prime(n+1)+Prime(k))/Prime(n) , k is the smallest 
number such that Prime(n+1)+Prime(k)==0  mod Prime(n) , n+1<k.

    %e A000008     a(1)=(Prime(2)+Prime(k))/Prime(1)=(3+5)/2=4


    Yasutoshi

 
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