The th
Ramanujan prime is the smallest number such that for all , where is the prime counting
function. In other words, there are at least primes between and whenever . The smallest such number must be prime, since the
function
can increase only at a prime.
Equivalently,
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. 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.Ramanujan, S. "A Proof of Bertrand's Postulate." J.
Indian Math. Soc.11, 181-182, 1919.Sloane, N. J. A.
Sequence A104272 in "The On-Line Encyclopedia
of Integer Sequences."
th
Ramanujan prime is the smallest number 
such that 
for all 
, where 
is the prime counting
function. In other words, there are at least 
primes between 
and 
whenever 
. The smallest such number 
must be prime, since the
function 
can increase only at a prime.
,
2, 3, 4, 5, ... if 
,
11, 17, 29, 41, ... (OEIS A104272), respectively.
These are the first few Ramanujan primes.
for all 
is Bertrand's postulate.
