A088790 Numbers k such that (k^k-1)/(k-1) is prime.

2, 3, 19, 31, 7547

Offset

1,1

Comments

Note that (k^k-1)/(k-1) is prime only if k is prime, in which case it equals cyclotomic(k,k), the k-th cyclotomic polynomial evaluated at x=k. This sequence is a subsequence of A070519. The number cyclotomic(7547,7547) is a probable prime found by H. Lifchitz. Are there only a finite number of these primes?

From T. D. Noe, Dec 16 2008: (Start)

The standard heuristic implies that there are an infinite number of these primes and that the next k should be between 10^10 and 10^11.

Let N = (7547^7547-1)/(7547-1) = A023037(7547). If N is prime, then the period of the Bell numbers modulo 7547 is N. See A054767. (End)

References

R. K. Guy, Unsolved Problems in Theory of Numbers, 1994, A3.

Links

Table of n, a(n) for n=1..5.

Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Mathematica

Do[p=Prime[n]; If[PrimeQ[(p^p-1)/(p-1)], Print[p]], {n, 100}]

Prog

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

Crossrefs

Cf. A070519 (cyclotomic(n, n) is prime).

Cf. A056826 ((n^n+1)/(n+1) is prime).

Sequence in context: A040145 A142955 A213896 * A283186 A363498 A215304

Adjacent sequences: A088787 A088788 A088789 * A088791 A088792 A088793

Keyword

hard,more,nonn

Author

T. D. Noe, Oct 16 2003

Status

approved