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<DIV><FONT face="MS UI Gothic">
<DIV><FONT face="MS UI Gothic"> These are prime related
sequences. </FONT></DIV>
<DIV><FONT face="MS UI Gothic"> I think the first one is
interesting. </FONT></DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> </FONT></DIV>
<DIV><FONT face="MS UI Gothic"> %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 <BR></FONT><FONT
face="MS UI Gothic"></FONT></DIV>
<DIV><FONT face="MS UI Gothic"> %N
A000001 a(n)=Prime(n)+Prime(n+k) , mod
4<BR>
k=1/2*(Prime(n+1)-Prime(n)) <BR> </FONT></DIV>
<DIV><FONT face="MS UI Gothic"> %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
.<BR>
For n=31, 61, 73, a(n)=2 <BR></FONT><FONT
face="MS UI Gothic"></FONT></DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT
face="MS UI Gothic">
I think the number of 2 is too few. I supposed that the </FONT><FONT
face="MS UI Gothic"> 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.<BR>
Does any other reason exist? </FONT></DIV>
<DIV> </DIV>
<DIV><FONT
face="MS UI Gothic">
I have no idea to calculate the exact ratio that limit k->infinity
{N(k)/(Pi(k)-Pi_2(k))} . </FONT></DIV>
<DIV><FONT
face="MS UI Gothic">
Where N(k) means number of terms of a(n)=2 up to k . N(k)=#{m | a(m)=2 ,
m<=k} . </FONT></DIV>
<DIV><FONT face="MS UI Gothic"> </FONT></DIV>
<DIV><FONT face="MS UI Gothic"> %0
A000001 2,10 ..... sequence starts from
n=2 </FONT></DIV>
<DIV><FONT face="MS UI Gothic"><BR> </DIV></FONT>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> %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
<BR> %N A000002
a(n)=Prime(n)+Prime(n+1) , mod 4<BR> %C
A000002 ratio=9/16 .....
understandable<BR> %O A000002 2,
3
</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face="MS UI Gothic">
</FONT></DIV>
<DIV><FONT face="MS UI Gothic"> %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 <BR> %N
A000003 a(n)=Prime(n-1)+Prime(n) , mod
4<BR>
k=1/2*(Prime(n+1)-Prime(n)) </FONT></DIV>
<DIV><FONT face="MS UI Gothic"> %O A000003
5,2 </FONT></DIV>
<DIV><FONT face="MS UI Gothic"> </DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> %S A000004 0, 0, 1, 1, 2,
1, 2, 2, 3, 2, 3, 4 <BR> %N
A000004 Number of ways of sum such that Prime(n)-1
=Prime(i)+Prime(j). <BR> %e
A000004
11-1=3+7=5+5 , so a(5)=2 . <BR> %O
A000004 1,5 </FONT></DIV>
<DIV> </DIV>
<DIV><FONT
face="MS UI Gothic"> <BR> %S
A000005 0, 0, 0, 1, 2, 2, 4, 2, 3, 4
</FONT></DIV>
<DIV><FONT face="MS UI Gothic"> %N
A000005 Number of ways of sum such that Prime(n) =
2*Prime(i)+Prime(j). </FONT></DIV>
<DIV><FONT face="MS UI Gothic"> %e
A000005
11=2*2+7=2*3+5 , so a(5)=2 </FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> %S A000006 2, 3, 11,
29</FONT><FONT face="MS UI Gothic">, 226 </FONT></DIV>
<DIV> %N A000006 a(n)=Sum( Product p_i , {Sum
p_i=Prime(n)} , p_i is prime ) </DIV>
<DIV> %e A000006
a(5)=2*2*2*2*3+2*2*2*5+2*2*7+2*3*3*3+3*3*5+11 </DIV>
<DIV> <FONT face="MS UI Gothic"><BR> </FONT></DIV>
<DIV><FONT face="MS UI Gothic"> %S A000007 1, 1, 2,
3, 6, 9,
17
<BR> %N A000007 Number of
partitions of a prime into primes.<BR> %e
A000007
a(5)=2+2+2+2+3=2+2+2+5=2+2+7=2+3+3+3=3+3+5=11 <FONT
face="MS UI Gothic"><BR></FONT> </FONT></DIV>
<DIV><FONT
face="MS UI Gothic"> <BR> %S
A000008 4, 4, 4, 4, 4, 6, 6, 4, 6, 12, 8, 4
<BR> %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.
<BR> <BR> %e
A000008 a(1)=(Prime(2)+Prime(k))/Prime(1)=(3+5)/2=4<BR></DIV></FONT>
<DIV><FONT
face="MS UI Gothic">
</FONT></DIV>
<DIV><FONT face="MS UI Gothic">
Yasutoshi </FONT></DIV>
<DIV><FONT face="MS UI Gothic"> </FONT></DIV>
<DIV> <FONT
face="MS UI Gothic"></DIV></FONT></FONT></FONT></DIV></BODY></HTML>