A309040 -id:A309040 - cp's OEIS frontend

A306825 Primitive part of A001353(n).

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

Offset: 1

Jianing Song, Mar 16 2019

nonn

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. [By the formula below, there is no such p. - Jianing Song, Oct 31 2024]

			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.

Eric Weisstein's World of Mathematics, Sylvester Cyclotomic Number
Wikipedia, Lucas Lehmer Primality Test

Cf. A001353, A238490, A309040.

Similar sequences: A061446, A008555.

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]))

Product_{d|n} a(d) = A001353(n), that is, a(n) = A001353(n)/(Product_{dA001353(d)^mu(n/d), where mu = A008683.

a(n) = A309040(2*n) for even n; A309040(n)*A309040(2*n) for odd n > 1. - Jianing Song, Oct 31 2024

A309526 a(n) is the greatest divisor of A001353(n) that is coprime to A001353(m) for all positive integers m < n.

Original entry on oeis.org

1, 4, 15, 7, 209, 13, 2911, 97, 901, 181, 564719, 193, 7865521, 2521, 6989, 18817, 1525870529, 2701, 21252634831, 37441, 6779137, 489061, 4122901604639, 37633, 274758906449, 6811741, 6575588101, 1037623, 11140078609864049, 40321, 155161278879431551

Offset: 1

Jianing Song, Aug 06 2019

nonn

Analog of A178763 and A308949.

Let b(n) = A309040(n)*gcd(A309040(n),n), then for n > 3: a(n) = b(2n) for even n and b(n)*b(2n) for odd n. It seems highly impossible that b(n) = 1 holds for n > 3, so it seems that only even-indexed terms can be primes.

			A001353(6) = 780 = 2^2 * 3 * 5 * 13. We have 2 divides A001353(2) = 2 and 3, 5 divides A001353(3) = 15, but A001353(m) is coprime to 13 for all 1 <= m < 6, so a(6) = 13.

Cf. A001353, A306825, A309040.

Cf. A178763, A308949, A309525.

T(n) = ([4, -1; 1, 0]^n)[2, 1]
b(n) = my(v=divisors(n)); prod(i=1, #v, T(v[i])^moebius(n/v[i]))
a(n) = if(isprime(n)&&!(12%n), b(n), b(n)/gcd(n, b(n)))

a(n) = A306825(n) / gcd(A306825(n), n) if n != 2, 3.

cp's OEIS Frontend

A306825 Primitive part of A001353(n).

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