# Prime related

Fri Mar 11 07:57:43 CET 2005

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

%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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