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A143223
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(Number of primes between n^2 and (n+1)^2) - (number of primes between n and 2n).
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10
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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
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OFFSET
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0,2
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COMMENTS
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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 < 30,000.
Contribution 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)
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REFERENCES
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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) 181-182.
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. [From Jonathan Sondow, Aug 02 2008]
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LINKS
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Table of n, a(n) for n=0..100.
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)
J. Pintz, Landau's problems on primes
J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld [From Jonathan Sondow, Aug 02 2008]
E. W. Weisstein, Legendre's Conjecture in MathWorld [From Jonathan Sondow, Aug 02 2008]
J. Sondow, Ramanujan Prime in MathWorld [From Jonathan Sondow, Aug 02 2008]
T. D. Noe, Plot of A143223(n) for n to 10^6 [From T. D. Noe, Aug 04 2008]
S. Ramanujan, A Proof Of Bertrand's Postulate [From Jonathan Sondow, Aug 02 2008]
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FORMULA
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a(n) = A014085(n) - A060715(n) (for n > 0) = [pi((n+1)^2) - pi(n^2)] - [pi(2n) - pi(n)] (for n > 1)
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EXAMPLE
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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. [From Jonathan Sondow, Aug 07 2008]
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MATHEMATICA
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L={0, 2}; Do[L=Append[L, (PrimePi[(n+1)^2]-PrimePi[n^2]) - (PrimePi[2n]-PrimePi[n])], {n, 2, 100}]; L
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CROSSREFS
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See A000720, A014085, A060715, A143224, A143225, A143226.
Negative terms are A143227. Cf. A104272 (Ramanujan primes). [From Jonathan Sondow, Aug 02 2008]
Sequence in context: A083896 A084115 A080028 * A063993 A115722 A115721
Adjacent sequences: A143220 A143221 A143222 * A143224 A143225 A143226
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KEYWORD
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sign
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AUTHOR
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Jonathan Sondow, Jul 31 2008
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EXTENSIONS
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Corrected by Jonathan Sondow, Aug 07 2008, Aug 09 2008
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STATUS
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approved
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