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

%I A133857 #38 Feb 16 2025 08:33:06
%S A133857 2,25667,28807,142031,157051,180181,414269,1270141
%N A133857 Numbers k such that (18^k - 1)/17 is prime.
%C A133857 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 A133857 With the discovery of a(6), this sequence of base-18 repunits is converging nicely to a rate close to Euler's constant with G=0.6667. - _Paul Bourdelais_, Mar 17 2010
%C A133857 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 A133857 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 A133857 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 A133857 Henri and Renaud Lifchitz, <a href="http://www.primenumbers.net/prptop/searchform.php?form=%2818%5Ex-1%29%2F17&amp;action=Search">PRP Records</a>.
%H A133857 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Repunit.html">Repunit</a>.
%e A133857 a(1) = A084740(18) = 2,
%e A133857 a(2) = A128164(18) = 25667.
%o A133857 (PARI) is(n)=ispseudoprime((18^n-1)/17) \\ _Charles R Greathouse IV_, Jun 13 2017
%Y A133857 Cf. A128164 (least k>2 such that (n^k-1)/(n-1) is prime).
%Y A133857 Cf. A084740 (least k such that (n^k-1)/(n-1) is prime).
%Y A133857 Cf. A126589 (numbers n>1 such that prime of the form (n^k-1)/(n-1) does not exist for k>2).
%K A133857 hard,more,nonn
%O A133857 1,1
%A A133857 _Alexander Adamchuk_, Sep 28 2007
%E A133857 a(2) = 25667 and a(3) = 28807 found by _Henri Lifchitz_, Sep 2007
%E A133857 a(4) corresponds to a probable prime discovered by _Paul Bourdelais_, Mar 12 2010
%E A133857 a(5) corresponds to a probable prime discovered by _Paul Bourdelais_, Mar 15 2010
%E A133857 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 A133857 a(7) corresponds to a probable prime discovered by _Paul Bourdelais_, Dec 08 2014
%E A133857 a(8) from _Paul Bourdelais_, Dec 02 2021