A014085 Number of primes between n^2 and (n+1)^2.

0, 2, 2, 2, 3, 2, 4, 3, 4, 3, 5, 4, 5, 5, 4, 6, 7, 5, 6, 6, 7, 7, 7, 6, 9, 8, 7, 8, 9, 8, 8, 10, 9, 10, 9, 10, 9, 9, 12, 11, 12, 11, 9, 12, 11, 13, 10, 13, 15, 10, 11, 15, 16, 12, 13, 11, 12, 17, 13, 16, 16, 13, 17, 15, 14, 16, 15, 15, 17, 13, 21, 15, 15, 17, 17, 18, 22, 14, 18, 23, 13

Offset

0,2

Comments

Suggested by Legendre's conjecture (still open) that for n > 0 there is always a prime between n^2 and (n+1)^2.

a(n) is the number of occurrences of n in A000006. - Philippe Deléham, Dec 17 2003

See the additional references and links mentioned in A143227. - Jonathan Sondow, Aug 03 2008

Legendre's conjecture may be written pi((n+1)^2) - pi(n^2) > 0 for all positive n, where pi(n) = A000720(n), [the prime counting function]. - Jonathan Vos Post, Jul 30 2008 [Comment corrected by Jonathan Sondow, Aug 15 2008]

Legendre's conjecture can be generalized as follows: for all integers n > 0 and all real numbers k > K, there is a prime in the range n^k to (n+1)^k. The constant K is conjectured to be log(127)/log(16). See A143935. - T. D. Noe, Sep 05 2008

For n > 0: number of occurrences of n^2 in A145445. - Reinhard Zumkeller, Jul 25 2014

References

J. R. Goldman, The Queen of Mathematics, 1998, p. 82.

Links

T. D. Noe, Table of n, a(n) for n = 0..10000

Pierre Dusart, The k-th prime is greater than k(ln k + ln ln k-1) for k>=2, Mathematics of Computation 68: (1999), 411-415.

Tsutomu 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.

Edmund Landau, Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion. Jahresbericht der Deutschen Mathematiker-Vereinigung (1912), Vol. 21, page 208-228.

Peter Munn, Logarithmic plot: number of primes between consecutive squares vs number of integers between the same squares

Michael Penn, Legendre's Conjecture is probably true, and here's why, YouTube video, 2023.

Hugo Pfoertner, Lower limit of the scatter band represented as a step function.

Eric Weisstein's World of Mathematics, Legendre's Conjecture

Wikipedia, Legendre's conjecture

Formula

a(n) = A000720((n+1)^2) - A000720(n^2). - Jonathan Vos Post, Jul 30 2008

a(n) = Sum_{k = n^2..(n+1)^2} A010051(k). - Reinhard Zumkeller, Mar 18 2012

Example

a(17) = 5 because between 17^2 and 18^2, i.e., 289 and 324, there are 5 primes (which are 293, 307, 311, 313, 317).

Mathematica

Table[PrimePi[(n + 1)^2] - PrimePi[n^2], {n, 0, 80}] (* Lei Zhou, Dec 01 2005 *)
Differences[PrimePi[Range[0, 90]^2]] (* Harvey P. Dale, Nov 25 2015 *)

Prog

(PARI) a(n)=primepi((n+1)^2)-primepi(n^2) \\ Charles R Greathouse IV, Jun 15 2011
(Haskell)
a014085 n = sum $ map a010051 [n^2..(n+1)^2]
-- Reinhard Zumkeller, Mar 18 2012
(Python)
from sympy import primepi
def a(n): return primepi((n+1)**2) - primepi(n**2)
print([a(n) for n in range(81)]) # Michael S. Branicky, Jul 05 2021

Crossrefs

First differences of A038107.

Cf. A000006, A053000, A053001, A007491, A077766, A077767, A108954, A000720, A060715, A104272, A143223, A143224, A143225, A143226, A143227.

Cf. A010051, A061265, A221056, A000290, A145445.

Counts of primes between consecutive higher powers: A060199, A061235, A062517.

Cf. A333846, A349996, A349997, A349998, A349999.

Sequence in context: A352631 A134446 A125749 * A248891 A171239 A029210

Adjacent sequences: A014082 A014083 A014084 * A014086 A014087 A014088

Keyword

nonn,nice

Author

Jon Wild, Jul 14 1997

Status

approved