A066180 a(n) = smallest base b so that repunit (b^prime(n) - 1) / (b - 1) is prime, where prime(n) = n-th prime; or 0 if no such base exists.

2, 2, 2, 2, 5, 2, 2, 2, 10, 6, 2, 61, 14, 15, 5, 24, 19, 2, 46, 3, 11, 22, 41, 2, 12, 22, 3, 2, 12, 86, 2, 7, 13, 11, 5, 29, 56, 30, 44, 60, 304, 5, 74, 118, 33, 156, 46, 183, 72, 606, 602, 223, 115, 37, 52, 104, 41, 6, 338, 217, 13, 136, 220, 162, 35, 10, 218, 19, 26, 39

Offset: 1

Author

Frank Ellermann, Dec 15 2001

Keywords

• nonn

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.)

Examples

			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.
		

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.

Crossrefs

Cf. A004023 (prime repunits in base 10), A000043 (prime repunits in base 2, Mersenne primes), A055129 (table of repunits).

Cf. A084732, A085398.

Programs

Mathematica

Table[p = Prime[n]; b = 1; While[b++; ! PrimeQ[(b^p - 1)/(b - 1)]]; b, {n, 1, 70}] (* Lei Zhou, Oct 07 2011 *)

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,","));

Formula

a(n) = A085398(prime(n)).

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