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Ramanujan primes: bounds, runs, twins, and gaps. (English) Zbl 1229.11014

The \(n\)th Ramanujan prime is the smallest positive integer \(R_n\) such that if \( x \geq R_n\), then the interval \((\frac{1}{2}x,x]\) contains at least \( n\) primes. The authors prove that the maximum of \( R_n/p_{3n}\) is \( R_5/p_{15} = 41/47\). They present statistics on the length of the longest run of Ramanujan primes among all primes \( p<10^n\), for \( n\leq9\). If an upper twin prime is Ramanujan, then so is the lower. Runs of Ramanujan primes are related to prime gaps. An appendix explains Noe’s fast algorithm for computing \(R_1,R_2,\dots ,R_n\).

MSC:

11A41 Primes

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Online Encyclopedia of Integer Sequences:

Lesser of twin primes.
Number of twin prime pairs below 10^n.
Ramanujan primes R_n: a(n) is the smallest number such that if x >= a(n), then pi(x) - pi(x/2) >= n, where pi(x) is the number of primes <= x.
Members of A164368 which are not Ramanujan primes.
Primes prime(k) such that all integers in the interval [(prime(k-1)+1)/2, (prime(k)-1)/2] are composite numbers.
The smaller members of twin prime pairs which are non-Ramanujan primes.
Ramanujan primes A104272(n) for which A104272(n) = A080359(n).
Size of the range of the Ramanujan Prime Corollary, 2*A168421(n) - A104272(n).
Isolated primes: Primes p such that there is no other prime in the interval [2*prevprime(p/2), 2*nextprime(p/2)].
Large Associated Ramanujan Prime, p_i.
Number of twin prime pairs < 10^n that contain at least one Ramanujan prime (A104272).
Prime numbers that are not Ramanujan primes.
Least Ramanujan prime beginning a run of n Ramanujan primes associated with a sharp prime gap.
Lesser of twin Ramanujan primes.
Lesser of twin primes if it is a Ramanujan prime.
Number of primes up to the n-th Ramanujan prime: A000720(A104272(n)).
Number of twin Ramanujan prime pairs less than 10^n.
Lengths of runs of consecutive isolated primes beginning with A166251(n).
Length of the longest run of Ramanujan primes that are consecutive primes < 10^n.
Length of the longest run of consecutive primes < 10^n that are not Ramanujan primes.
Decimal expansion of Ramanujan prime constant: Sum_{n>=1} (1/R_n)^2, where R_n is the n-th Ramanujan prime, A104272(n).
Decimal expansion of sum of alternating series of reciprocals of Ramanujan primes, Sum_{n>=1} (1/R_n)(-1)^(n-1), where R_n is the n-th Ramanujan prime, A104272(n).
primepi(R_{n*m}) <= n*primepi(R_m) for m >= a(n), where R_k is the k-th Ramanujan prime (A104272).
primepi(R_m) <= i*primepi(R_j) for any factorization m=i*j if j >= a(i), where R_k is the k-th Ramanujan prime (A104272).
First differences of A179196, pi(R_(n+1)) - pi(R_n) where R_n is A104272(n).
Number of Ramanujan primes R_k between triangular numbers T(n-1) < R_k <= T(n).
Last known occurrence of number n of Ramanujan primes in A191225.
Greatest Ramanujan prime index less than x, eta(x).
Ramanujan primes of the second kind: a(n) is the smallest prime such that if prime x >= a(n), then pi(x) - pi(x/2) >= n, where pi(x) is the number of primes <= x.
a(n) is the maximal prime, such that for all primes x<=a(n) the number of primes in (x/2,x) is less than n.
a(n) is the smallest prime(m) such that the interval (prime(m)*n, prime(m+1)*n) contains exactly one prime.
Even numbers that are not the sum of two non-Ramanujan primes (A174635).
Number of decompositions of 2n into an unordered sum of two non-Ramanujan primes (A174635).
Last occurrence of n partitions in A205617.
Let p_n=prime(n), n>=1. Then a(n) is the least prime p which differs from p_n, for which the intervals (p/2,p_n/2), (p,p_n], if p<p_n, or the intervals (p_n/2,p/2), (p_n,p], if p>p_n, contain the same number of primes, and a(n)=0, if no such prime p exists.
a(n) is the smallest n-isolated prime, or a(n)=0 if there are no n-isolated primes.
Let (p(n), p(n)+2) be the n-th twin prime pair. a(n) is the smallest k, such that there is only one prime in the interval (k*p(n), k*(p(n)+2)), or a(n)=0, if there is no such k.
Let (p,p+2) be the n-th twin prime pair. a(n) is the least integer r > 1 for which the interval (r*p, r*(p+2)) contains no primes, or a(n)=0, if no such r exists.
R(n) - prime(2n), where R(n) is the n-th Ramanujan prime and prime(n) is the n-th prime.