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A066180
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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.
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13
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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
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1,1
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COMMENTS
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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.)
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REFERENCES
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Paulo Ribenboim, "The New Book of Prime Numbers Records", Springer, 1996, p. 353.
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LINKS
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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 (World of Mathematics).
H. C. Williams & E. Seah, Some primes of the form: (a^n - 1)/(a - 1), Mathematics of Computation 23, 1979.
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FORMULA
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a(n) = A085398(prime(n)).
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EXAMPLE
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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.
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MATHEMATICA
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Table[p = Prime[n]; b = 1; While[b++; ! PrimeQ[(b^p - 1)/(b - 1)]]; b, {n, 1, 70}] (* Lei Zhou, Oct 07 2011 *)
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PROG
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(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, ", "));
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CROSSREFS
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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: A337228 A341444 A084731 * A123487 A130325 A154097
Adjacent sequences: A066177 A066178 A066179 * A066181 A066182 A066183
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KEYWORD
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nonn
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AUTHOR
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Frank Ellermann, Dec 15 2001
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EXTENSIONS
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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
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STATUS
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approved
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