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<DIV><FONT face="MS UI Gothic" size=2> Neil wrote :
</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> >The OEIS will be on
holiday for the rest of the year!<BR></FONT><FONT face="MS UI Gothic"
size=2> >But only very important new sequences and Comments
should be submitted.<BR></DIV></FONT>
<DIV><FONT face="MS UI Gothic" size=2> I don't know if it is
important or not.</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> If not, ignore
it.</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> I will post it again
next year.</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> %I
A000001<BR> %S A000001 3, 35, 75361<BR> %N
A000001 a(n) = (Product_{0<=e_i<=1} (Product_{1<=i<=n}
p_i^e_i + Product_{1<=i<=n} p_i^(1-e_i)))^(1/2)
<BR>
Where p_i means i-th prime.<BR> %C A000001 This is a "Proof of
existence of infinite primes"
sequence.<BR>
Proof. Let N = (Product_{0<=e_i<=1} (Product_{1<=i<=n} p_i^e_i
+ Product_{1<=i<=n} p_i^(1-e_i)))^(1/2) . Suppose there are only a finite
number of primes p_i, 1<=i<=n. If N is prime, then for all i, not (N=p_i).
Because, for all i, p_i<N. If N is composite, then it must have a prime
divisor p which is different from primes p_i. Because, for all i, not (N_1=0,
Mod p_i).</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2> %e A000001
a(3)=
((1+p_1*p_2*p_3)*(p_3+p_1*p_2)*(p_2+p_1*p_3)*(p_2*p_3+p_1)*(p_1+p_2*p_3)*(p_1*p_3+p_2)*(p_1*p_2+p_3)*(p_1*p_2*p_3+1))^(1/2)<BR>
=
(1+p_1*p_2*p_3)*(p_3+p_1*p_2)*(p_2+p_1*p_3)*(p_2*p_3+p_1)<BR>
= 31*11*13*17 </FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2> %Y A000001
A111392 <BR> %K A000001
none<BR> %O A000001 1,1<BR> %A A000001
Yasutsohi Kohmoto <A
href="mailto:zbi74583@boat.zero.ad.jp">zbi74583@boat.zero.ad.jp</A>
</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2> I am not sure if the
representation on %N line is correct.<BR> Is it
understandable?<BR> </FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> Yasutoshi</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2>
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