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 A306825 Primitive part of A001353(n). 2
 1, 4, 15, 14, 209, 13, 2911, 194, 2703, 181, 564719, 193, 7865521, 2521, 34945, 37634, 1525870529, 2701, 21252634831, 37441, 6779137, 489061, 4122901604639, 37633, 274758906449, 6811741, 19726764303, 7263361, 11140078609864049, 40321, 155161278879431551 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS A prime p is called a unique-period prime in base b if there is no other prime q such that the period length of 1/q is equal to that of 1/p. If q = a(2p) = A001353(2*p)/(4*A001353(p)) = ((2 + sqrt(3))^p + (2 - sqrt(3))^p)/4 is prime (this happens for p = 3, 5, 7, 11, 13, 17, 19, 79, 151, 199, 233, 251, 317, ...), where p is an odd prime, then q is a unique-period prime in base b = (sqrt(12*q^2 - 3) - 1)/2 (1/q has period length 3) as well as in base b' = (sqrt(12*q^2 - 3) + 1)/2 (1/q has period length 6). For example, a(6) = 13 is prime, so 13 is the only prime whose reciprocal has period length 3 in base 22 and the only prime whose reciprocal has period length 6 in base 23. Compare: If q = A000129(p) = A008555(p), then q is a unique-period prime in base b = sqrt(2*q^2 - 1) (1/q has period length 4). By Lucas-Lehmer test, p is a Mersenne prime > 3 if and only if the smallest k such that p divides a(k) is k = (p - 1)/2. For primes p, p^2 divides a(k) for some k if and only if p = 2 or p is in A238490. If p > 2, the only possible values for k are the divisors of (p - Legendre(3,p))/2 (e.g., 103^2 divides a(52) = 53028360515521 = 103^2 * 4998431569). Conjecturally there must be infinitely many primes p such that a(p) is prime, but no such p is known. LINKS Eric Weisstein's World of Mathematics, Sylvester Cyclotomic Number Wikipedia, Lucas Lehmer Primality Test FORMULA Product_{d|n} a(d) = A001353(n), that is, a(n) = A001353(n)/(Product_{d

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Last modified January 27 10:39 EST 2022. Contains 350607 sequences. (Running on oeis4.)