login
This site is supported by donations to The OEIS Foundation.

 

Logo

Many excellent designs for a new banner were submitted. We will use the best of them in rotation.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A143225 Number of primes between n^2 and (n+1)^2, if equal to the number of primes between n and 2n. 9
0, 3, 9, 9, 10, 10, 16, 20, 19, 21, 23, 23, 24, 25, 28, 31, 32, 36, 38, 56, 57, 59, 59, 62, 65, 71, 75, 84, 88, 88, 96, 102, 107, 115, 116, 119, 120, 126, 125, 129, 132, 132, 163, 168, 168, 182, 189, 189, 192, 197, 198, 213, 236 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Legendre's conjecture (still open) says there is always a prime between n^2 and (n+1)^2. Bertrand's postulate (actually a theorem due to Chebyshev) says there is always a prime between n and 2n.

See the additional reference and link to Ramanujan's work mentioned in A143223. [From Jonathan Sondow, Aug 03 2008]

REFERENCES

M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer, NY, 2001.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Oxford Univ. Press, 1989, p. 19.

S. Ramanujan, "A Proof of Bertrand's Postulate," J. Indian Math. Soc. 11 (1919) 181-182.

LINKS

T. D. Noe, Table of n, a(n) for n=1..97 (no other n < 10^6)

T. Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate

M. Hassani, Counting primes in the interval (n^2,(n+1)^2)

J. Pintz, Landau's problems on primes

J. Sondow, Ramanujan Prime in MathWorld [From Jonathan Sondow, Aug 02 2008]

J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld [From Jonathan Sondow, Aug 02 2008]

E. W. Weisstein, Legendre's Conjecture in MathWorld [From Jonathan Sondow, Aug 02 2008]

FORMULA

a(n) = A014085(A143224(n)) = A060715(A143224(n)) for n > 0

EXAMPLE

There are 3 primes between 9^2 and 10^2 and 3 primes between 9 and 2*9, so 3 is a member.

MATHEMATICA

L={}; Do[If[PrimePi[(n+1)^2]-PrimePi[n^2] == PrimePi[2n]-PrimePi[n], L=Append[L, PrimePi[2n]-PrimePi[n]]], {n, 0, 2000}]; L

CROSSREFS

See A000720, A014085, A060715, A143223, A143224, A143226.

Cf. A104272, A143227. [From Jonathan Sondow, Aug 03 2008]

Sequence in context: A004166 A110759 A063750 * A223195 A203600 A099720

Adjacent sequences:  A143222 A143223 A143224 * A143226 A143227 A143228

KEYWORD

nonn

AUTHOR

Jonathan Sondow, Jul 31 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified April 18 23:11 EDT 2014. Contains 240734 sequences.