Using simple properties of the gamma function, Ramanujan (1919) gave a new proof of Bertrand's
postulate. Then he proved the generalization that , 2, 3, 4, 5, ... if , 11, 17, 29, 41, ... (OEIS A104272),
respectively. These are the first few Ramanujan primes.
Ramanujan, S. "A Proof of Bertrand's Postulate." J. Indian Math. Soc.11, 181-182, 1919.Ramanujan, S. Collected
Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, P. V. S. Aiyar,
and B. M. Wilson). Providence, RI: Amer. Math. Soc., pp. 208-209,
2000.Sondow, J. "Ramanujan Primes and Bertrand's Postulate."
Amer. Math. Monthly116, 630-635, 2009. https://doi.org/10.1080/00029890.2009.11920980.Sloane,
N. J. A. Sequence A104272 in "The
On-Line Encyclopedia of Integer Sequences."
th
Ramanujan prime is the smallest positive integer 
such that 
for all real
numbers 
,
where 
is a positive integer and 
is the prime counting
function (Sondow 2009). In other words, there are at least 
primes in the interval ![(x/2,x]](/images/equations/RamanujanPrime/Inline8.svg)
whenever 
. The smallest such number 
must be prime, since the
function 
can increase only at a prime.
ranges over the positive integers.
, 2, 3, 4, 5, ... if 
, 11, 17, 29, 41, ... (OEIS A104272),
respectively. These are the first few Ramanujan primes.
for all 
is an equivalent form of Bertrand's postulate.
