OFFSET
1,1
COMMENTS
Is a(n) = 0 possible?
Let p be the n-th prime; Cp(x) be the p-th cyclotomic polynomial (x^p - 1)/(x - 1); a(n) is the least k > 1 such that Cp(k) is prime.
The values associated with a(5) and a(8) through a(70) have been certified prime with Primo. (a(1) through a(4), a(6) and a(7) give prime(2), prime(4), prime(11), prime(31), prime(1028) and prime(12251), respectively.)
REFERENCES
Paulo Ribenboim, "The New Book of Prime Numbers Records", Springer, 1996, p. 353.
LINKS
Robert G. Wilson v, Table of n, a(n) for n = 1..300 (terms 1..200 from Charles R Greathouse IV).
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
Andy Steward, Titanic Prime Generalized Repunits.
Eric Weisstein's World of Mathematics, Repunit.
H. C. Williams and E. Seah, Some primes of the form: (a^n - 1)/(a - 1), Mathematics of Computation 23, 1979.
FORMULA
a(n) = A085398(prime(n)).
EXAMPLE
a(5) = 5 because 11 is the 5th prime; (b^5 - 1)/(b - 1) is composite for b = 2,3,4 and prime ((5^11 - 1)/4 = 12207031) for b = 5.
b = 61 for prime(12) = 37 because (61^37 - 1)/60 is prime and 61 is the least base b that makes (b^37 - 1)/(b - 1) a prime.
MATHEMATICA
Table[p = Prime[n]; b = 1; While[b++; ! PrimeQ[(b^p - 1)/(b - 1)]]; b, {n, 1, 70}] (* Lei Zhou, Oct 07 2011 *)
PROG
(PARI) /* This program assumes (probable) primes exist for each n. */
/* All 70 (probable) primes found by this program have been proved prime. */
gen_repunit(b, n) = (b^prime(n)-1)/(b-1);
for(n=1, 70, b=1; until(isprime(p), b++; p=gen_repunit(b, n)); print1(b, ", "));
CROSSREFS
KEYWORD
nonn
AUTHOR
Frank Ellermann, Dec 15 2001
EXTENSIONS
Sequence extended to 16 terms by Don Reble, Dec 18 2001
More terms from Rick L. Shepherd, Sep 14 2002
Entry revised by N. J. A. Sloane, Jul 23 2006
STATUS
approved