Prime related sequences

[kohmoto]

    Neil
    Sorry for the delay.
    My condition was not so good.
    Have I written you a mail that I am in a rehabilitation of a sickness of 
mind?

    Yasutoshi


     %I A000001
     %S A000001     2, 1, 2, 1, 5, 3, 8, 4, 3, 14, 5, 9
    %N A000001    [Prime(n)/(Prime(n+1)-Prime(n))] , where [x] means the 
integer part of x.
　　%O A000001    1, 1
     %K A000001    nonn
     %A A000001    Yasutoshi Kohmoto (zbi74583 at boat.zero.ad.jp)

    %I A000002
    %S A000002    0, 1, 1, 1, 1, 1, 1, 2, 2,
    %V A000002    0, -1, 1, -1, 1, -1, -1, 2, -2,
    %N A000002    Prime(n)-(Prime(n+1)+Prime(n-1))/2
    %C A000002    a(n)=-1/2*(A001223(n+1)-A001223(n))
    %O A000002    5, 8
    %K A000002    sign
    %A A000002    Yasutoshi Kohmoto (zbi74583 at boat.zero.ad.jp)

    %I A000003
    %S A000003    0, 1, 2, 1, 2, 2, 4, 3, 2, 2, 1,
    %N A000003    a(n)={minimal k such that f^k (Prime(n))=1}
                        where f(m)=(m+1)/2^r , 2^r is the highest power of 
two dividing m+1.
    %e A000003 f(f(f(f(17))))=1 , Prime(7)=17, so a(7)=4
    %O A000003    1, 3
    %K A000003    nonn
    %A A000003    Yasutoshi Kohmoto (zbi74583 at boat.zero.ad.jp)

    %I A000004
    %S A000004    0, 2, 1, 3, 2, 1, 1, 2, 3, 1, 5, 1,
    %N A000004    2^a(n) is the largest power of two dividing (Prime(n)+1)
    %O A000004    1, 2
    %K A000003    nonn
    %A A000003    Yasutoshi Kohmoto (zbi74583 at boat.zero.ad.jp)

    %I A000005
     %S A000005    1, 7, 4, 31, 31, 28, 50, 127, 91, 217, 122, 124
    %N A000005    Sum {|d|^2 , d|n} .
                        where |d| means norm of d. d|n means d divides n.
                        If norms of two divisors of n are same, then they 
are counted as the same number.
    %e A000005    a(2)=1+|1+i|^2+2^2=7 ,
                        a(5)=1+|1+2i|^2+5^2=31, |1+2i|=|2+i| so they are 
counted as one divisor.
　　%O A000005    1, 2
     %K A000005    nonn
     %A A000005    Yasutoshi Kohmoto (zbi74583 at boat.zero.ad.jp)

    This is the first line of an array a(n,k) : Sum{|d|^2 , d|(n+k*i)} , k=0
     Do you need the sequences for k=1, 2, 3, 4 ...?
 
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