OEIS sequence A299145 lists the primes of the form Q(n,k)=∑ni=2ik for n≥2 and k>0. It is clear that except for the case k=1 and n=2 resulting in the prime 2, we must have n≥3.
The sum Q(n,k) is equal to H(n,−k)−1 where H(n,m)=∑ni=11im is the generalized harmonic number of order n of m. It is well known that H(n,−k) is a polynomial of n of degree k+1. In particular, Faulhaber's formula shows that
H(n,−k)=12nk+1k+1⌊k/2⌋∑j=0(k+12j)B2jnk+1−2j
where Bi is the i-th Bernoulli number.
H(n,k) and thus Q(n,k) can be written as a degree k+1 polynomial of n with rational coefficients. The smallest denominator of this polynomial is found in OEIS A064538. In 2017, Kellner and Sondow showed that this is equal to (k+1)∏p∈S where S is the set of primes p≤k+22+(kmod2) such that the sum of the base p digits of k+1 is p or larger.
Since H(1,−k)=1, this implies that Q(1,k)=0 and thus n−1 is a factor of Q(n,k). Since n≥3, if n−1 does not get cancelled out by a factor of the denominator, we would have a nontrivial factor of Q(n,k) and thus Q(n,k) is not prime. This implies that Q(n,k) is prime only if n−1 is a divisor of a(k) in sequence A064538. Thus for each k there is only a finite number of values of n to check. This provides an efficient algorithm to find terms of this sequence by looking only for primes in the numbers Q(n,k) where n−1 is a divisor of A064538(k). There are 45 such numbers with 1000 or less digits.
References
Bernd C. Kellner, Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly 124 (2017), 695-709.
Subscribe to:
Post Comments (Atom)
No comments:
Post a Comment