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<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></BODY></HTML>