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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

Table of n, a(n) for n=1..31.

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

EXAMPLE

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.

PROG

(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)

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

CROSSREFS

Cf. A001353, A238490.

Similar sequences: A061446, A008555.

Sequence in context: A332767 A095331 A163773 * A210788 A229412 A076349

Adjacent sequences:  A306822 A306823 A306824 * A306826 A306827 A306828

KEYWORD

nonn

AUTHOR

Jianing Song, Mar 16 2019

STATUS

approved

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