

A014085


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


114



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
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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.
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


REFERENCES

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


LINKS



FORMULA

Conjecture: for all n>1, abs(a(n)(n/log(n))) < sqrt(n).  Alain Rocchelli, Sep 20 2023


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

(Haskell)
a014085 n = sum $ map a010051 [n^2..(n+1)^2]
(Python)
from sympy import primepi
def a(n): return primepi((n+1)**2)  primepi(n**2)


CROSSREFS

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


KEYWORD

nonn,nice


AUTHOR



STATUS

approved



