A143223 (Number of primes between n^2 and (n+1)^2) - (number of primes between n and 2n).

0, 2, 1, 1, 1, 1, 2, 1, 2, 0, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 3, 2, 1, 1, 3, 2, 1, 1, 2, 2, 1, 3, 2, 3, 1, 2, 0, 0, 3, 2, 2, 2, -1, 3, 2, 3, 0, 4, 6, 0, 1, 4, 4, 1, 1, -2, -1, 3, -1, 3, 3, 1, 5, 3, 1, 3, 1, 2, 4, -1, 6, 1, 1, 4, 4, 4, 7, -1, 3, 8, -2, 5, 3, 5, 1, 0, 5, 5, 1, 2, 3, 2, 1, 5, 3, 3, 2, 3, 4, 1, 2

Offset

0,2

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.

Hashimoto's plot of (1 - a(n)) shows that |a(n)| is small compared to n for n < 30000.

From Jonathan Sondow, Aug 07 2008: (Start)

It appears that there are only a finite number of negative terms (see A143226).

If the negative terms are bounded, then Legendre's conjecture is true, at least for all sufficiently large n. This follows from the strong form of Bertrand's postulate proved by Ramanujan (see A104272 Ramanujan primes). (End)

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, Collected Papers of Srinivasa Ramanujan (G. H. Hardy, S. Aiyar, P. Venkatesvara and B. M. Wilson, eds.), Amer. Math. Soc., Providence, 2000, pp. 208-209.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

T. Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate, arXiv:0807.3690 [math.GM], 2008.

M. Hassani, Counting primes in the interval (n^2,(n+1)^2), arXiv:math/0607096 [math.NT], 2006.

T. D. Noe, Plot of A143223(n) for n to 10^6

J. Pintz, Landau's problems on primes

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

J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld

J. Sondow, Ramanujan Prime in MathWorld

E. W. Weisstein, Legendre's Conjecture in MathWorld

Formula

a(n) = A014085(n) - A060715(n) (for n > 0) = [pi((n+1)^2) - pi(n^2)] - [pi(2n) - pi(n)] (for n > 1).

Example

There are 4 primes between 6^2 and 7^2 and 2 primes between 6 and 2*6, so a(6) = 4 - 2 = 2.
a(1) = 2 because there are two primes between 1^2 and 2^2 (namely, 2 and 3) and none between 1 and 2. [Jonathan Sondow, Aug 07 2008]

Mathematica

L={0, 2}; Do[L=Append[L, (PrimePi[(n+1)^2]-PrimePi[n^2]) - (PrimePi[2n]-PrimePi[n])], {n, 2, 100}]; L

Prog

(PARI) a(n)=sum(k=n^2+1, n^2+2*n, isprime(k))-sum(k=n+1, 2*n, isprime(k)) \\ Charles R Greathouse IV, May 30 2014

Crossrefs

See A000720, A014085, A060715, A143224, A143225, A143226.

Negative terms are A143227. Cf. A104272 (Ramanujan primes).

Sequence in context: A080028 A309228 A309778 * A063993 A353445 A115722

Adjacent sequences: A143220 A143221 A143222 * A143224 A143225 A143226

Keyword

sign

Author

Jonathan Sondow, Jul 31 2008

Extensions

Corrected by Jonathan Sondow, Aug 07 2008, Aug 09 2008

Status

approved