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A143226 Numbers n such that there are more primes between n and 2n than between n^2 and (n+1)^2. 10
42, 55, 56, 58, 69, 77, 80, 119, 136, 137, 143, 145, 149, 156, 174, 177, 178, 188, 219, 225, 232, 247, 253, 254, 257, 261, 263, 297, 306, 310, 325, 327, 331, 335, 339, 341, 344, 356, 379, 395, 402, 410, 418, 421, 425, 433, 451, 485, 500 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
It appears that this sequence is finite; searching up to 10^5, the last n appears to be 48717. [T. D. Noe, Aug 01 2008]
If the sequence is finite, then, by Bertrand's postulate, Legendre's conjecture is true for sufficiently large n. - Jonathan Sondow, Aug 02 2008
No other n <= 10^6. The plot of A143223 shows that it is quite likely that there are no additional terms. - T. D. Noe, Aug 04 2008
See the additional reference and link to Ramanujan's work mentioned in A143223. - Jonathan Sondow, Aug 03 2008
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.
LINKS
M. Hassani, Counting primes in the interval (n^2,(n+1)^2), arXiv:math/0607096 [math.NT], 2006.
S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld
FORMULA
A143223(n) < 0.
EXAMPLE
There are 10 primes between 42 and 2*42, but only 9 primes between 42^2 and 43^2, so 42 is a member.
MATHEMATICA
L={}; Do[If[PrimePi[(n+1)^2]-PrimePi[n^2] < PrimePi[2n]-PrimePi[n], L=Append[L, n]], {n, 0, 500}]; L
CROSSREFS
Sequence in context: A182147 A029695 A307986 * A043136 A039313 A043916
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Jul 31 2008
STATUS
approved

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Last modified July 13 21:38 EDT 2024. Contains 374288 sequences. (Running on oeis4.)