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

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

Offset: 1

Author

Alexander Adamchuk, Sep 28 2007

Keywords

• hard
• more
• nonn

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

Examples

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

Links

• 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.

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).

Programs

PARI

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

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