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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.

%I A066180 #59 Feb 16 2025 08:32:45
%S A066180 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 A066180 12,86,2,7,13,11,5,29,56,30,44,60,304,5,74,118,33,156,46,183,72,606,
%U A066180 602,223,115,37,52,104,41,6,338,217,13,136,220,162,35,10,218,19,26,39
%N 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.
%C A066180 Is a(n) = 0 possible?
%C A066180 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 A066180 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 A066180 Paulo Ribenboim, "The New Book of Prime Numbers Records", Springer, 1996, p. 353.
%H A066180 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 A066180 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 A066180 Andy Steward, <a href="http://www.primes.viner-steward.org/andy/titans.html">Titanic Prime Generalized Repunits</a>.
%H A066180 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Repunit.html">Repunit</a>.
%H A066180 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 A066180 a(n) = A085398(prime(n)).
%e A066180 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 A066180 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 A066180 Table[p = Prime[n]; b = 1; While[b++; ! PrimeQ[(b^p - 1)/(b - 1)]]; b, {n, 1, 70}] (* _Lei Zhou_, Oct 07 2011 *)
%o A066180 (PARI) /* This program assumes (probable) primes exist for each n. */
%o A066180 /* All 70 (probable) primes found by this program have been proved prime. */
%o A066180 gen_repunit(b,n) = (b^prime(n)-1)/(b-1);
%o A066180 for(n=1,70, b=1; until(isprime(p), b++; p=gen_repunit(b,n)); print1(b,","));
%Y A066180 Cf. A004023 (prime repunits in base 10), A000043 (prime repunits in base 2, Mersenne primes), A055129 (table of repunits).
%Y A066180 Cf. A084732, A085398.
%K A066180 nonn
%O A066180 1,1
%A A066180 _Frank Ellermann_, Dec 15 2001
%E A066180 Sequence extended to 16 terms by _Don Reble_, Dec 18 2001
%E A066180 More terms from _Rick L. Shepherd_, Sep 14 2002
%E A066180 Entry revised by _N. J. A. Sloane_, Jul 23 2006