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

2, 25667, 28807, 142031, 157051, 180181, 414269, 1270141

Offset

1,1

Comments

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

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

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

Links

Table of n, a(n) for n=1..8.

Paul Bourdelais, Generalized Repunit Conjecture - Paul Bourdelais, Mar 12 2010

H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.

Henri and Renaud Lifchitz, PRP Records.

Eric Weisstein's World of Mathematics, Repunit.

Example

a(1) = A084740(18) = 2,
a(2) = A128164(18) = 25667.

Prog

(PARI) is(n)=ispseudoprime((18^n-1)/17) \\ Charles R Greathouse IV, Jun 13 2017

Crossrefs

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).

Sequence in context: A132942 A237521 A131558 * A349519 A232869 A153912

Adjacent sequences: A133854 A133855 A133856 * A133858 A133859 A133860

Keyword

hard,more,nonn

Author

Alexander Adamchuk, Sep 28 2007

Extensions

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

a(4) corresponds to a probable prime discovered by Paul Bourdelais, Mar 12 2010

a(5) corresponds to a probable prime discovered by Paul Bourdelais, Mar 15 2010

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

a(7) corresponds to a probable prime discovered by Paul Bourdelais, Dec 08 2014

a(8) from Paul Bourdelais, Dec 02 2021

Status

approved