1,1

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.).

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

H. Dubner, Generalized repunit primes, Math. of Comput. 61, 1993

Paulo Ribenboim ."The New Book of Prime Numbers Records" Springer 1996 Page 353

Williams, H. C. & Seah, E., Some primes of the form: (a^n - 1)/ (a - 1), Mathematics of Computation 23, 1979.

Charles R Greathouse IV, Table of n, a(n) for n = 1..200

Andy Steward, Prime Generalized Repunits.

Eric Weisstein's World of Mathematics, Repunit (World of Mathematics).

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 p(12)=37 because (61^37 - 1)/36 = prime and 61 is the least base that makes (b^37 -1)/36 a prime.

(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, ", "))

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

Cf. A084732, A085398.

Sequence in context: A103512 A130086 A084731 this_sequence A123487 A130325 A154097

Adjacent sequences: A066177 A066178 A066179 this_sequence A066181 A066182 A066183

nonn

Frank.Ellermann(AT)t-online.de, Dec 15 2001

Sequence extended to 16 terms by Don Reble (djr(AT)nk.ca), Dec 18 2001.

More terms from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 14 2002

Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Jul 23 2006