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<DIV><FONT size=2>There is theory very close to what you are doing, but not
exactly.</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>For odd n, let the sequence R_n(k) = n*2^k-1 for k >=
1. For example, R_13 is the</FONT></DIV>
<DIV><FONT size=2>sequence</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2></FONT><FONT size=2> 25, 51, 103, 207, 415,
831, ...</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>n is called a Riesel number if no primes occur in its Riesel
sequence. 13 is not</FONT></DIV>
<DIV><FONT size=2>a Riesel number because 103 shows up in its Riesel
sequence.</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>For any n, R_n satisfies the recurrence</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2> R_n(k+1) = 2*R_n(k)+1,</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>That is, precisely the relationship you are
investigating.</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>Every odd number belongs to a unique R_n. Specifically,
your starting number 73</FONT></DIV>
<DIV><FONT size=2>belongs to R_37, which starts</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2> 73, 147, 295, 591, 1183, 2368, ...</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>That is R_37 is precisely the sequence you are
considering. However, 37 is not</FONT></DIV>
<DIV><FONT size=2>a Riesel number precisely because 73 is prime.</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>You saw the "obvious pattern" in the divisors of R_37.
Specifically, prime divisors</FONT></DIV>
<DIV><FONT size=2>show up periodically in the sequence. For example, 3
divides R_37(2k), and</FONT></DIV>
<DIV><FONT size=2>5 divides R_37(4k+3), etc. This is true for any prime
that divides any element of</FONT></DIV>
<DIV><FONT size=2>a Riesel sequence.</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>In fact, the way that we prove that a number is Riesel is by
showing that there is</FONT></DIV>
<DIV><FONT size=2>a periodic sequence of prime divisors. The first proven
Riesel number is 509203.</FONT></DIV>
<DIV><FONT size=2>R_509203 looks like</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2> 1018405, 2036811, 4073623, 8147247, 16294495,
...</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>It has a cyclic pattern of divisors:</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2> 5, 3, 241, 3, 5, 3, 13, 3, 5, 3, 7, 3, 5, 3, 17,
3, 5, 3, 13, 3, 5, 3, 7, 3, ...</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>where these 24 divisors repeat forever. This means that
every element of R_509203</FONT></DIV>
<DIV><FONT size=2>is (properly) divisible by 3, 5, 7, 13, 17 or 241, and is
therefore composite. So 509203</FONT></DIV>
<DIV><FONT size=2>is Riesel.</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>If R_n does not have a cyclical pattern of divisors, a
probabilistic argument indicates</FONT></DIV>
<DIV><FONT size=2>that R_n should have a prime element, and n should not be
Riesel. However, this is</FONT></DIV>
<DIV><FONT size=2>not a proof that n is not Riesel, to do this we must prove
there is a prime in R_n, and</FONT></DIV>
<DIV><FONT size=2>the only way we know to do this is to find and exhibit
that prime.</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>n = 509203 is the smallest proven Riesel number; it has
long been conjectured that it</FONT></DIV>
<DIV><FONT size=2>is in fact the smallest Riesel number. To prove this, it
is necessary to show that all</FONT></DIV>
<DIV><FONT size=2>n < 509203 are non-Riesel. Unfortunately, there are a
few stubborn n < 509203 for</FONT></DIV>
<DIV><FONT size=2>which neither a cyclic pattern of divisors of R_n or a prime
element of R_n has been</FONT></DIV>
<DIV><FONT size=2>established. Eliminating these Riesel candidates is the
goal of the Riesel Sieve Project</FONT></DIV>
<DIV><FONT size=2>(<A
href="http://www.rieselsieve.com">www.rieselsieve.com</A>). You can go
there for more background information.</FONT></DIV>
<DIV><FONT size=2>(note, more Riesel numbers are given in A101036).</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>To sum up:</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>The kind of sequence you are investigating has been looked at
before. For this type</FONT></DIV>
<DIV><FONT size=2>of sequence, you can either show it contains no primes by
finding a cyclic pattern of</FONT></DIV>
<DIV><FONT size=2>divisors, or show it does contain a prime by finding it.
For some numbers, finding the</FONT></DIV>
<DIV><FONT size=2>prime is very difficult. You are lucky that Don Reble
came to your assistance in this</FONT></DIV>
<DIV><FONT size=2>instance (I will assume that he is correct that 74*2552-1 is
prime).</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>There are much more difficult examples than the one you
gave, though. For example,</FONT></DIV>
<DIV><FONT size=2>R_2293 =</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2> 4585, 9171, 18343, 36687, 73375,
...</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>(see <A
href="http://www.prothsearch.net/rieselsearch.html">http://www.prothsearch.net/rieselsearch.html</A>).
</FONT><FONT size=2>So problems such as yours are</FONT></DIV>
<DIV><FONT size=2>known and </FONT><FONT size=2>heavily
investigated.</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2> </FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV>----- Original Message ----- </DIV>
<BLOCKQUOTE dir=ltr
style="PADDING-RIGHT: 0px; PADDING-LEFT: 5px; MARGIN-LEFT: 5px; BORDER-LEFT: #000000 2px solid; MARGIN-RIGHT: 0px">
<DIV
style="BACKGROUND: #e4e4e4; FONT: 10pt arial; font-color: black"><B>From:</B>
<A title=zakirs@yosh.ac.il href="mailto:zakirs@yosh.ac.il">זקיר סעידוב -
ד\"ר/Zakir Seidov Ph.D.</A> </DIV>
<DIV style="FONT: 10pt arial"><B>To:</B> <A title=ham
href="mailto:ham">ham</A> ; <A title=seqfan@ext.jussieu.fr
href="mailto:seqfan@ext.jussieu.fr">seqfan@ext.jussieu.fr</A> </DIV>
<DIV style="FONT: 10pt arial"><B>Sent:</B> Friday, March 11, 2005 6:29
AM</DIV>
<DIV style="FONT: 10pt arial"><B>Subject:</B> n => 2n+1 to get prime: seed
= 73</DIV>
<DIV><BR></DIV>
<P><FONT size=2>Dear Seqfans,<BR></FONT><FONT size=2>The operation n =>
2n+1 quickly gives primes for most "seed" values of n.<BR></FONT><FONT
size=2>But for some seeds, the transformed numbers keep being
composite.<BR></FONT><FONT size=2>The first "tough" number is n=73.<BR>Here is
the list of least divisors of transformed numbers<BR></FONT><FONT size=2>(I do
not consider "seed" itself which in this case happened to be prime
):<BR></FONT><FONT size=2>{3,5,3,7,</FONT><FONT
size=2><BR>3,5,3,19,</FONT><FONT size=2><BR>3,5,3,47,</FONT><FONT
size=2><BR>3,5,3,7,</FONT><FONT size=2><BR>3,5,3,61,</FONT><FONT
size=2><BR>3,5,3,29,<BR></FONT><FONT size=2>3,5,3,7,<BR></FONT><FONT
size=2>3,5,3,1439,<BR></FONT><FONT size=2>3,5,3,73,<BR></FONT><FONT
size=2>3,5,3,7,<BR></FONT><FONT size=2>3,5,3,19,<BR></FONT><FONT
size=2>3,5,3,46703,<BR></FONT><FONT size=2>3,5,3,7,<BR></FONT><FONT
size=2>3,5,3,22247,<BR></FONT><FONT size=2>3,5,3,59,<BR></FONT><FONT
size=2>3,5,3,7,<BR></FONT><FONT size=2>3,5,3,761,<BR></FONT><FONT
size=2>3,5,3,73,<BR></FONT><FONT size=2>3,5,3,7,<BR></FONT><FONT
size=2>3,5,3,19,<BR></FONT><FONT size=2>3,5,3,137,<BR></FONT><FONT
size=2>3,5,3,7,<BR></FONT><FONT size=2>3,5,3,131381,<BR></FONT><FONT
size=2>3,5,3,2411639,<BR></FONT><FONT size=2>3,5,3,7}.<BR></FONT><FONT
size=2>The clear pattern is seen, which may help to search the prime
case.<BR></FONT><FONT size=2>My request is:<BR></FONT><FONT
size=2>Can the n =>2n+1 transformation, in this particular
case,<BR> lead to prime number (and when?),<BR></FONT><FONT size=2>may
anyone bother to find it?<BR></FONT><FONT size=2>What about general
theory?</FONT></P>
<P align=left><FONT size=2>Thank you very much,<BR></FONT><FONT
size=2>Zak</FONT></P>
<P align=left><FONT size=2>PS</FONT><FONT size=2> The last considered
number, <BR>93806144416888975710756037197823,<BR></FONT><FONT
size=2>has the full list of divisors as follows:<BR></FONT><FONT
size=2>{{7, 1}, {13, 1}, {599, 1}, {21214924397, 1}, {81118812239619751, 1}}
-<BR></FONT><FONT size=2>at least according to Mathematica.<BR></FONT><FONT
size=2>And next three numbers are clearly composite according the pattern
mentioned.</FONT></P></BLOCKQUOTE></BODY></HTML>