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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. 9
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 (list; graph; refs; listen; history; text; internal format)
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.).

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

REFERENCES

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.

LINKS

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

Andy Steward, Titanic Prime Generalized Repunits.

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

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

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 * A123487 A130325 A154097

Adjacent sequences:  A066177 A066178 A066179 * A066181 A066182 A066183

KEYWORD

nonn,changed

AUTHOR

Frank Ellermann, Dec 15 2001

EXTENSIONS

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

More terms from Rick L. Shepherd, Sep 14 2002

Entry revised by N. J. A. Sloane, Jul 23 2006

STATUS

approved

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Last modified November 24 01:09 EST 2014. Contains 249867 sequences.