

A143227


(Number of primes between n and 2n)  (number of primes between n^2 and (n+1)^2), if > 0.


11



1, 2, 1, 1, 1, 1, 2, 2, 3, 3, 3, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1, 2, 2, 6, 3, 3, 1, 1, 1, 2, 1, 1, 1, 1, 6, 3, 8, 3, 2, 3, 2, 3, 1, 1, 4, 3, 10, 2, 1, 1, 2, 3, 1, 3, 4, 2, 2, 9, 7, 2, 2, 4, 3, 3, 1, 2, 3, 5, 1, 2, 3, 2, 11, 3, 1, 2, 4, 7, 1, 1, 1, 1, 1, 5, 1, 2, 3, 3, 4, 2, 2, 9, 5, 1, 4, 2, 2
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OFFSET

1,2


COMMENTS

If the sequence is bounded (e.g., if it is finite), then Legendre's conjecture is true: there is always a prime between n^2 and (n+1)^2, at least for all sufficiently large n. This follows from the strong form of Bertrand's postulate proved by Ramanujan (see A104272 Ramanujan primes).


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) 181182.
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. 208209.


LINKS

T. D. Noe, Table of n, a(n) for n=1..413
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)
T. D. Noe, Plot of the points (A143226(n), A143227(n))
J. Pintz, Landau's problems on primes
J. Sondow, Ramanujan Prime in MathWorld
J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld
E. W. Weisstein, Legendre's Conjecture in MathWorld


FORMULA

a(n) = A143223(A143226(n))


EXAMPLE

The first positive value of ((pi(2n)  pi(n))  (pi((n+1)^2)  pi(n^2))) is 1 (at n = 42), the 2nd is 2 (at n = 55) and the 3rd is 1 (at n = 56), so a(1) = 1, a(2) = 2, a(3) = 1.


MATHEMATICA

L={}; Do[ With[ {d=(PrimePi[2n]PrimePi[n])(PrimePi[(n+1)^2]PrimePi[n^2])}, If[d>0, L=Append[L, d]]], {n, 0, 1000}]; L


CROSSREFS

Cf. A000720, A014085, A060715, A104272, A143223, A143224, A143225, A143226 = corresponding values of n.
Sequence in context: A064823 A140225 A104758 * A026791 A080576 A213211
Adjacent sequences: A143224 A143225 A143226 * A143228 A143229 A143230


KEYWORD

nonn


AUTHOR

Jonathan Sondow, Aug 02 2008


STATUS

approved



