

A066180


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


12



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
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OFFSET

1,1


COMMENTS

Is a(n) = 0 possible?
Let p be the nth prime; Cp(x) be the pth 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), 927930.
Andy Steward, Titanic Prime Generalized Repunits.
Eric Weisstein's World of Mathematics, Repunit (World of Mathematics).
H. C. Williams & 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)/(b1);
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


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



