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A133857 Numbers n such that (18^n - 1)/17 is prime. 1


%S 2,25667,28807,142031,157051,180181,414269

%N Numbers n such that (18^n - 1)/17 is prime.

%C Repunits in base 18 are off to a slow start compared with all the repunits in bases from -20 to 20. There are only 4 repunit primes in base 18 with exponents searched up to 150,000 while most other bases have 7-10 by then. Even after scaling the rate by logb logb, this is relatively low. - _Paul Bourdelais_, Mar 12 2010

%C With the discovery of a(6), this sequence in base 18 repunits is converging nicely to a rate close to Euler's constant with G=0.6667. - _Paul Bourdelais_, Mar 17 2010

%C With the discovery of a(7), G=0.54789, which is very close to the expected constant 0.56145948 mentioned in the Generalized Repunit Conjecture below. - _Paul Bourdelais_, Dec 08 2014

%H Paul Bourdelais,<a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906">Generalized Repunit Conjecture</a> - _Paul Bourdelais_, Mar 12 2010

%H H. Dubner, <a href="http://dx.doi.org/10.1090/S0025-5718-1993-1185243-9">Generalized repunit primes</a>, Math. Comp., 61 (1993), 927-930.

%H Henri & Renaud Lifchitz, <a href="http://www.primenumbers.net/prptop/searchform.php?form=%2818%5Ex-1%29%2F17&amp;action=Search">PRP Records</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repunit.html">Repunit</a>.

%e a(1) = A084740(18) = 2,

%e a(2) = A128164(18) = 25667.

%o (PARI) is(n)=ispseudoprime((18^n-1)/17) \\ _Charles R Greathouse IV_, Jun 13 2017

%Y Cf. A128164 (Least k>2 such that (n^k-1)/(n-1) is prime). Cf. A084740 (Least k such that (n^k-1)/(n-1) is prime). Cf. A126589 (Numbers n>1 such that prime of the form (n^k-1)/(n-1) does not exist for k>2).

%K hard,more,nonn

%O 1,1

%A _Alexander Adamchuk_, Sep 28 2007

%E a(2) = 25667 and a(3) = 28807 found by Henri Lifchitz, Sep 2007

%E a(4) corresponds to a probable prime discovered by _Paul Bourdelais_, Mar 12 2010

%E a(5) corresponds to a probable prime discovered by _Paul Bourdelais_, Mar 15 2010

%E a(6)=180181, previously discovered by Andy Steward in April 2007 in the form of the cyclotomic number Phi(180181,18), added by _Paul Bourdelais_, Mar 23 2010

%E a(7) corresponds to a probable prime discovered by _Paul Bourdelais_, Dec 08 2014

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Last modified June 4 08:18 EDT 2020. Contains 334825 sequences. (Running on oeis4.)