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A014085 Number of primes between n^2 and (n+1)^2. 83
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 (list; graph; refs; listen; history; text; internal format)
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.

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

Tsutomu 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)

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

Wikipedia, Legendre's conjecture

FORMULA

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

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

a(n) = sum (A010051(k) : k = n^2 .. (n+1)^2). [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*)

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

CROSSREFS

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

Cf. A010051.

Cf. A061265, A038107, A221056.

Cf. A000290, A145445.

Sequence in context: A126336 A134446 A125749 * A248891 A171239 A029210

Adjacent sequences:  A014082 A014083 A014084 * A014086 A014087 A014088

KEYWORD

nonn,nice

AUTHOR

Jon Wild

STATUS

approved

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Last modified December 17 21:13 EST 2014. Contains 252040 sequences.