A306825 - OEIS

%I #35 Oct 31 2024 21:55:46

%S 1,4,15,14,209,13,2911,194,2703,181,564719,193,7865521,2521,34945,

%T 37634,1525870529,2701,21252634831,37441,6779137,489061,4122901604639,

%U 37633,274758906449,6811741,19726764303,7263361,11140078609864049,40321,155161278879431551

%N Primitive part of A001353(n).

%C 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).

%C 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.

%C 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).

%C Conjecturally there must be infinitely many primes p such that a(p) is prime, but no such p is known. [By the formula below, there is no such p. - _Jianing Song_, Oct 31 2024]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SylvesterCyclotomicNumber.html">Sylvester Cyclotomic Number</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas-Lehmer_primality_test">Lucas Lehmer Primality Test</a>

%F Product_{d|n} a(d) = A001353(n), that is, a(n) = A001353(n)/(Product_{d<n, d|n} a(d)). Equivalently, a(n) = Product_{d|n} A001353(d)^mu(n/d), where mu = A008683.

%F a(n) = A309040(2*n) for even n; A309040(n)*A309040(2*n) for odd n > 1. - _Jianing Song_, Oct 31 2024

%e For n = 8 we have: a(1) = A001353(1), a(1)*a(2) = A001353(2), a(1)*a(2)*a(4) = A001353(4), a(1)*a(2)*a(4)*a(8) = A001353(8). The solution is a(1) = 1, a(2) = 4, a(4) = 14 and a(8) = 194.

%o (PARI) b(n) = if(n==1, [1], my(v=vector(n)); v[1]=1; v[2]=4; for(i=3, n, v[i]=4*v[i-1]-v[i-2]); v)

%o a(n) = my(d=divisors(n)); prod(i=1, #d, (b(n)[d[i]])^moebius(n/d[i]))

%Y Cf. A001353, A238490, A309040.

%Y Similar sequences: A061446, A008555.

%K nonn

%O 1,2

%A _Jianing Song_, Mar 16 2019