cp's OEIS Frontend

The Ramanujan primes possess the following property:

If p = prime(n) > 2, then all numbers (p+1)/2, (p+3)/2, ..., (prime(n+1)-1)/2 are composite.

The sequence equals all primes with this property, whether Ramanujan or not.

All Ramanujan primes A104272 are in the sequence.

Every lesser of twin primes (A001359), beginning with 11, is in the sequence. - Vladimir Shevelev, Aug 31 2009

109 is the first non-Ramanujan prime in this sequence.

A very simple sieve for the generation of the terms is the following: let p_0=1 and, for n>=1, p_n be the n-th prime. Consider consecutive intervals of the form (2p_n, 2p_{n+1}), n=0,1,2,... From every interval containing at least one prime we remove the last one. Then all remaining primes form the sequence. Let us demonstrate this sieve: For p_n=1,2,3,5,7,11,... consider intervals (2,4), (4,6), (6,10), (10,14), (14,22), (22,26), (26,34), ... . Removing from the set of all primes the last prime of each interval, i.e., 3,5,7,13,19,23,31,... we obtain 2,11,17,29, etc. - Vladimir Shevelev, Aug 30 2011

This sequence and A194598 are the mutually wrapping up sequences:

A194598(1) <= a(1) <= A194598(2) <= a(2) <= ...

From Peter Munn, Oct 30 2017: (Start)

The sequence is the list of primes p = prime(k) such that there are no primes between prime(k)/2 and prime(k+1)/2. Changing "k" to "k-1" and therefore "k+1" to "k" produces a definition very similar to A164333's: it differs by prefixing an initial term 3. From this we get a(n+1) = prevprime(A164333(n)) = A151799(A164333(n)) for n >= 1.

The sequence is the list of primes that are not the largest prime less than 2*prime(k) for any k, so that - as a set - it is the complement relative to A000040 of the set of numbers in A059788.

{{2}, A166252, A166307} is a partition.

(End)

Number of primes between successive even semiprimes. [Juri-Stepan Gerasimov, May 01 2010]

From Peter Munn, Jun 01 2023: (Start)

First differences of A020900.

A080192 lists prime(n) corresponding to the zero terms.

A104380(k) is prime(n) corresponding to the first occurrence of k as a term.

If a(n) is nonzero, A059786(n) is the smallest and A059788(n+1) the largest of the a(n) enumerated primes. In the tree of primes described in A290183, these primes label the child nodes of prime(n).

Conjecture: the asymptotic proportions of 0's, 1's, ... , k's, ... are 1/3, 2/9, ... , 2^k/3^(k+1), ... .

(End)

Complement of A104272 in the primes. Not the same as A059788.

Also known as non-Ramanujan Primes. - John W. Nicholson, Jan 29 2012

Starting from some prime, iterated application of A060308 (or of the equivalent A059788) generates a chain of increasing prime numbers.

The nature of these chains is to reach higher in the list of primes, sometimes "over-satisfying" Bertrand's postulate by skipping some nearer primes, almost doubling of possible. On the other hand, A164368 contains the primes that would be skipped by a chain which contains the prime slightly above half of their value. The sequence shows how far up in chains starting from some member of A164368 we find prime(n), or equivalently, how many inverse applications of the map we need to hit a member of A164368 if starting at prime(n).

Note that by construction A164368(k) starts with the smallest prime that is not member of any chain started from any previous A164368. So each prime exists at some place in one of these chains, and the number of steps a(n) to reach it from the start of its chain is well defined.

Complement of A080192 relative to A000040.

From Flávio V. Fernandes, May 26 2021: (Start)

Equivalently, primes p such that p is the largest of all prime factors of the numbers in the interval [2*p, nextprime(2*p)-1].

For any prime p, if p is not the largest of all prime factors of the numbers in that interval (i.e., if p is not a term of this sequence), then the largest of all prime factors of the numbers in that interval will be a prime q that occurs in the number 2*q.

For all n, the largest prime < 2*a(n) is a term of A059788. (End)

The parent of the node labeled p is the adjacent node through which it is connected to the root.

The "tree of primes" defined above relates to many older sequences. The node labeled A055377(n) is parent of the node labeled n. The node labeled prime(k) has A102820(k) child nodes and unless it has no child nodes, these are labeled with the primes from A059786(k) to A059788(k+1). The leaf node labels are A080192. The nodes of depth m are those with labels in the interval [A055496(m), A055496(m+1)). The full tree may be defined using A000040 read as a table with row lengths given by A102820 prefixed by 2.

If the set of heights of nodes has a greatest finite value, k, this sequence is finite with k+1 contiguous defined terms.

That said, the author's initial assessment is that occurrence of height n nodes will have similarities to occurrence of least primes of prime k-tuples, namely: (1) labels of nodes of height n will occur almost as though at random intervals amongst the primes; (2) for any n, the apparent odds against a prime p being such a label will not be greater than polynomial in log(p); and thus (3) a(n) plausibly exists for all n.

Some initial empirical observation suggests nodes of height n+1 may occur something like 5 to 10 times less frequently than those of height n.

Terms a(1) to a(5) come from the subtree consisting of the node labeled 17 and its descendants, as depicted in the example section below. This implies 4 consecutive negative first differences, which may be rare later in the sequence.