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%I #58 Feb 21 2023 09:44:17
%S 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,
%T 12,86,2,7,13,11,5,29,56,30,44,60,304,5,74,118,33,156,46,183,72,606,
%U 602,223,115,37,52,104,41,6,338,217,13,136,220,162,35,10,218,19,26,39
%N 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.
%C Is a(n) = 0 possible?
%C 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.
%C 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.)
%D Paulo Ribenboim, "The New Book of Prime Numbers Records", Springer, 1996, p. 353.
%H Robert G. Wilson v, <a href="/A066180/b066180.txt">Table of n, a(n) for n = 1..300</a> (terms 1..200 from Charles R Greathouse IV).
%H H. Dubner, <a href="http://dx.doi.org/10.1090/S0025-5718-1993-1185243-9">Generalized repunit primes</a>, Math. Comp., 61 (1993), 927-930.
%H Andy Steward, <a href="http://www.primes.viner-steward.org/andy/titans.html">Titanic Prime Generalized Repunits</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repunit.html">Repunit</a>.
%H H. C. Williams and E. Seah, <a href="http://dx.doi.org/10.1090/S0025-5718-1979-0537980-7">Some primes of the form: (a^n - 1)/(a - 1)</a>, Mathematics of Computation 23, 1979.
%F a(n) = A085398(prime(n)).
%e 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.
%e 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.
%t Table[p = Prime[n]; b = 1; While[b++; ! PrimeQ[(b^p - 1)/(b - 1)]]; b, {n, 1, 70}] (* _Lei Zhou_, Oct 07 2011 *)
%o (PARI) /* This program assumes (probable) primes exist for each n. */
%o /* All 70 (probable) primes found by this program have been proved prime. */
%o gen_repunit(b,n) = (b^prime(n)-1)/(b-1);
%o for(n=1,70, b=1; until(isprime(p), b++; p=gen_repunit(b,n)); print1(b,","));
%Y Cf. A004023 (prime repunits in base 10), A000043 (prime repunits in base 2, Mersenne primes), A055129 (table of repunits).
%Y Cf. A084732, A085398.
%K nonn
%O 1,1
%A _Frank Ellermann_, Dec 15 2001
%E Sequence extended to 16 terms by _Don Reble_, Dec 18 2001
%E More terms from _Rick L. Shepherd_, Sep 14 2002
%E Entry revised by _N. J. A. Sloane_, Jul 23 2006
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