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300 BCE Euclid proved in his work Elements (Book IX, Proposition 20) that there is an infinite number of primes; that is, there is no greatest prime. In 1845, Joseph Bertrand made the observation that in any interval there is always at least one prime. In 1919, Ramanujan refined Bertrand's observation by proving there is a steady (but not monotonic) increase in the tally of primes in the intervals as increases. Ramanujan said even more: while the tally of primes jitters around a bit—as seen by the rough contour of this plot—you can always identify that point in the integers where going forward the tally of primes is never less than some value. That point is always a prime, and collectively these points are known as the Ramanujan primes.

The On-line Encyclopedia of Integer Sequences (OEIS) lists the Ramanujan primes at [1]. The sequence starts 2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167, 179, 181, ….

About 46% of primes less than 19,000 are Ramanujan primes.

About 15% of primes less than 19,000 are the lesser of twin primes.

About 78% of the lesser of twin primes less than 19,000 are Ramanujan primes.

About 26% of the Ramanujan primes less than 19,000 are the lesser of twin primes.

Snapshots 1 and 2: Highlight the first two consecutive Ramanujan primes, and , which are also consecutive regular primes, and . Integer sequence A189993 in the OEIS lists the longest runs of consecutive regular primes that are Ramanujan primes up to for .

Snapshot 3: primes 73, 79, 83, 89 all fail to be , which is a privilege reserved for

References

[1] J. Sondow. The On-Line Encyclopedia of Integer Sequences. (Apr 19 2023) oeis.org/A104272.

[2] J. Sondow, J. W. Nicholson and T. D. Noe, "Ramanujan Primes: Bounds, Runs, Twins, and Gaps," Journal of Integer Sequences, 14, 2011 Article 11.6.2. cs.uwaterloo.ca/journals/JIS/VOL14/Noe/noe12.html.