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A143225 Number of primes between n^2 and (n+1)^2, if equal to the number of primes between n and 2n. 10
0, 3, 9, 9, 10, 10, 16, 20, 19, 21, 23, 23, 24, 25, 28, 31, 32, 36, 38, 56, 57, 59, 59, 62, 65, 71, 75, 84, 88, 88, 96, 102, 107, 115, 116, 119, 120, 126, 125, 129, 132, 132, 163, 168, 168, 182, 189, 189, 192, 197, 198, 213, 236 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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.
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
T. D. Noe, Table of n, a(n) for n=1..97 (no other n < 10^6)
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
a(n) = A014085(A143224(n)) = A060715(A143224(n)) for n > 0.
EXAMPLE
There are 3 primes between 9^2 and 10^2 and 3 primes between 9 and 2*9, so 3 is a member.
MATHEMATICA
L={}; Do[If[PrimePi[(n+1)^2]-PrimePi[n^2] == PrimePi[2n]-PrimePi[n], L=Append[L, PrimePi[2n]-PrimePi[n]]], {n, 0, 2000}]; L
CROSSREFS
Cf. A104272, A143227. [Jonathan Sondow, Aug 03 2008]
Sequence in context: A004166 A110759 A063750 * A223195 A203600 A099720
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Jul 31 2008
STATUS
approved

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Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)