A014085 -id:A014085 - cp's OEIS frontend

A089610 Number of primes between n^2 and (n+1/2)^2.

1, 1, 1, 2, 1, 2, 1, 2, 2, 4, 2, 2, 3, 2, 4, 4, 1, 2, 3, 3, 4, 4, 2, 4, 4, 4, 4, 4, 4, 4, 5, 5, 6, 4, 5, 7, 3, 6, 6, 8, 5, 5, 7, 4, 6, 7, 6, 7, 6, 6, 5, 9, 7, 7, 6, 7, 7, 6, 8, 8, 7, 7, 8, 9, 11, 7, 8, 10, 8, 11, 8, 7, 7, 10, 11, 12, 4, 9, 11, 6, 9, 9, 10, 8, 9, 8, 11, 8, 8, 9, 10, 8, 13, 10, 9, 10, 14, 12

Offset: 1

Cino Hilliard, Dec 30 2003

For small values of n, these numbers exhibit higher and lower values as n increases. Conjectures: After n=17 a(n) > 1. There exists an n_1 such that a(n) is < a(n+1) for all n >= n_1.

Same as the number of primes between n^2 and n^2+n. Oppermann conjectured in 1882 that a(n)>0. - T. D. Noe, Sep 16 2008

Paulo Ribenboim, The New Book of Prime Number Records, 3rd ed., 1995, Springer, p. 248.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 183.

T. D. Noe, Table of n, a(n) for n=1..10000
Wikipedia, Oppermann's conjecture

Cf. A010051, A014085, A094189, A108309.

a089610 n = sum $ map a010051' [n^2 .. n*(n+1)]
-- Reinhard Zumkeller, Jun 07 2015

a[n_] := PrimePi[(n + 1/2)^2] - PrimePi[n^2]; Table[ a@n, {n, 100}] (* Robert G. Wilson v, May 04 2009 *)

a(n) = primepi(n^2+n) - primepi(n^2); \\ Michel Marcus, May 18 2020

A143223 (Number of primes between n^2 and (n+1)^2) - (number of primes between n and 2n).

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow, Jul 31 2008

sign

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 < 30000.
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)

			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. [_Jonathan Sondow_, Aug 07 2008]

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, 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.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
T. Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate, arXiv:0807.3690 [math.GM], 2008.
M. Hassani, Counting primes in the interval (n^2,(n+1)^2), arXiv:math/0607096 [math.NT], 2006.
T. D. Noe, Plot of A143223(n) for n to 10^6
J. Pintz, Landau's problems on primes
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
J. Sondow, Ramanujan Prime in MathWorld
E. W. Weisstein, Legendre's Conjecture in MathWorld

See A000720, A014085, A060715, A143224, A143225, A143226.

Negative terms are A143227. Cf. A104272 (Ramanujan primes).

L={0,2}; Do[L=Append[L,(PrimePi[(n+1)^2]-PrimePi[n^2]) - (PrimePi[2n]-PrimePi[n])], {n,2,100}]; L

a(n)=sum(k=n^2+1,n^2+2*n,isprime(k))-sum(k=n+1,2*n,isprime(k)) \\ Charles R Greathouse IV, May 30 2014

a(n) = A014085(n) - A060715(n) (for n > 0) = [pi((n+1)^2) - pi(n^2)] - [pi(2n) - pi(n)] (for n > 1).

Corrected by Jonathan Sondow, Aug 07 2008, Aug 09 2008

A143224 Numbers n such that (number of primes between n^2 and (n+1)^2) = (number of primes between n and 2n).

Original entry on oeis.org

0, 9, 36, 37, 46, 49, 85, 102, 107, 118, 122, 127, 129, 140, 157, 184, 194, 216, 228, 360, 365, 377, 378, 406, 416, 487, 511, 571, 609, 614, 672, 733, 767, 806, 813, 863, 869, 916, 923, 950, 978, 988, 1249, 1279, 1280, 1385, 1427, 1437, 1483, 1539, 1551, 1690

Offset: 1

Jonathan Sondow, Jul 31 2008

nonn

The sequence gives the positions of zeros in A143223. The number of primes in question is A143225(n).

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.

			There is the same number of primes (namely 3) between 9^2 and 10^2 as between 9 and 2*9, so 9 is a term.

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, 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. [Jonathan Sondow, Aug 03 2008]

T. D. Noe, Table of n, a(n) for n=1..97 (no other n < 10^6)
T. Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate, arXiv:0807.3690 [math.GM], 2008.
M. Hassani, Counting primes in the interval (n^2,(n+1)^2), arXiv:math/0607096 [math.NT], 2006.
J. Pintz, Landau's problems on primes
S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
J. Sondow, Ramanujan Prime in MathWorld.
J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld.
Eric Weisstein's World of Mathematics, Legendre's Conjecture.

See A000720, A014085, A060715, A143223, A143225, A143226.

Cf. A104272, A143227. [Jonathan Sondow, Aug 03 2008]

with(numtheory): A143224:=n->`if`(pi((n+1)^2)-pi(n^2) = pi(2*n)-pi(n), n, NULL): seq(A143224(n), n=0..2000); # Wesley Ivan Hurt, Jul 25 2017

L={}; Do[If[PrimePi[(n+1)^2]-PrimePi[n^2] == PrimePi[2n]-PrimePi[n], L=Append[L,n]], {n,0,2000}]; L
(* Second program *)
With[{nn = 2000}, {0}~Join~Position[#, {0}][[All, 1]] &@ Map[Differences, Transpose@ {Differences@ Array[PrimePi[#^2] &, nn], Array[PrimePi[2 #] - PrimePi[#] &, nn - 1]}]] (* Michael De Vlieger, Jul 25 2017 *)

is(n) = primepi((n+1)^2)-primepi(n^2)==primepi(2*n)-primepi(n) \\ Felix Fröhlich, Jul 25 2017

A143223(a(n)) = 0.

A143225 Number of primes between n^2 and (n+1)^2, if equal to the number of primes between n and 2n.

Original entry on oeis.org

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

Offset: 1

Jonathan Sondow, Jul 31 2008

nonn

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]

			There are 3 primes between 9^2 and 10^2 and 3 primes between 9 and 2*9, so 3 is a member.

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.

T. D. Noe, Table of n, a(n) for n=1..97 (no other n < 10^6)
T. Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate, arXiv:0807.3690 [math.GM], 2008.
M. Hassani, Counting primes in the interval (n^2,(n+1)^2), arXiv:math/0607096 [math.NT], 2006.
J. Pintz, Landau's problems on primes
S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
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

See A000720, A014085, A060715, A143223, A143224, A143226.

Cf. A104272, A143227. [Jonathan Sondow, Aug 03 2008]

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

a(n) = A014085(A143224(n)) = A060715(A143224(n)) for n > 0.

A143226 Numbers n such that there are more primes between n and 2n than between n^2 and (n+1)^2.

Original entry on oeis.org

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

Offset: 1

Jonathan Sondow, Jul 31 2008

nonn

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

			There are 10 primes between 42 and 2*42, but only 9 primes between 42^2 and 43^2, so 42 is a member.

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.

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, arXiv:0807.3690 [math.GM], 2008.
M. Hassani, Counting primes in the interval (n^2,(n+1)^2), arXiv:math/0607096 [math.NT], 2006.
J. Pintz, Landau's problems on primes
S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
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

See A000720, A014085, A060715, A143223, A143224, A143225, A104272, A143227.

L={}; Do[If[PrimePi[(n+1)^2]-PrimePi[n^2] < PrimePi[2n]-PrimePi[n], L=Append[L,n]], {n,0,500}]; L

A143223(n) < 0.

A143935 Number of primes between n^K and (n+1)^K, inclusive, where K=log(127)/log(16).

Original entry on oeis.org

2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 5, 1, 4, 2, 3, 3, 4, 1, 5, 3, 3, 4, 3, 3, 3, 4, 4, 3, 5, 4, 3, 5, 2, 4, 5, 4, 5, 5, 3, 5, 5, 2, 6, 5, 4, 4, 4, 5, 5, 7, 5, 5, 3, 5, 6, 3, 8, 3, 4, 5, 6, 7, 5, 6, 8, 5, 4, 6, 6, 3, 7, 5, 4, 8, 5, 8, 6, 3, 7, 7, 6, 8, 7, 4, 5, 6, 5, 9, 9, 7, 6, 6, 6, 6, 7, 6, 4, 8, 5, 8, 8, 4

Offset: 1

T. D. Noe, Sep 05 2008

nonn

This value of K is conjectured to be the least possible such that there is at least one prime in the range n^k and (n+1)^k for all n>0 and k>=K. This value of K was found using exact interval arithmetic. For each n <= 300 and for each prime p in the range n to n^2, we computed an interval k(n,p) such that p is between n^k(n,p) and (n+1)^k(n,p). The intersection of all these intervals produces a list of 29 intervals. The last interval appears to be semi-infinite beginning with K, which is log(127)/log(16). See A143898 for the smallest number in the first interval.
My UBASIC program indicates no prime between 113.457 ... and 126.999 .... Next prime > 113 is 127. I would like someone to check this. - Enoch Haga, Sep 24 2008
It suffices to check members of floor(A002386^(1/k)). - Charles R Greathouse IV, Feb 03 2011
The constant log(127)/log(16) is A194361. - John W. Nicholson, Dec 13 2013

T. D. Noe, Table of n, a(n) for n = 1..10000
Carlos Rivera, Conjecture 60: Generalization of Legendre's Conjecture

Cf. A014085 (number of primes between n^2 and (n+1)^2).

Cf. A144256, A194361.

k= 1.74717117169304146332; Table[Length[Select[Range[Ceiling[n^k],Floor[(n+1)^k]], PrimeQ]], {n,150}]
With[{k=Log[16,127]},Table[Count[Range[Ceiling[n^k],Floor[(n+1)^k]],?PrimeQ],{n,110}]] (* _Harvey P. Dale, Apr 03 2019 *)

Corrected a(15) from 1 to 0 Enoch Haga, Sep 24 2008

My intention was to include the endpoints of the range. Using k=log(127)/log(16), the endpoint for n=15 is exactly 127, which is prime. - T. D. Noe, Sep 25 2008

A061265 Number of squares between n-th prime and (n+1)st prime.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1

Offset: 1

Amarnath Murthy, Apr 24 2001

If n-th prime is a member of A053001 then a(n) is at least 1. If not, then a(n) = 0.
Legendre's conjecture (still open) that there is always a prime between n^2 and (n+1)^2 is equivalent to conjecturing that a(n) <= 1 for all n. - Vladeta Jovovic, May 01 2003
a(A038107(n)) = 1 for n > 1; a(A221056(n)) = 0. - Reinhard Zumkeller, Apr 15 2013

			a(3) = 0 as there is no square between 5, the third prime and 7, the fourth prime. a(4) = 1, as there is a square (9) between the 4th prime 7 and the 5th prime 11.

Harry J. Smith, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Legendre's Conjecture
Wikipedia, Legendre's conjecture

Cf. A053001.
Cf. A038107.
Cf. A014085.

a061265 n = a061265_list !! (n-1)
a061265_list = map sum $
   zipWith (\u v -> map a010052 [u..v]) a000040_list $ tail a000040_list
-- Reinhard Zumkeller, Apr 15 2013

ns[{a_,b_}]:=Count[Range[a+1,b-1],?(IntegerQ[Sqrt[#]]&)]; ns/@ Partition[ Prime[Range[110]],2,1] (* _Harvey P. Dale, Mar 14 2015 *)

{ n=0; q=2; forprime (p=3, prime(2001), write("b061265.txt", n++, " ", floor(sqrt(p))-floor(sqrt(q))); q=p ) } \\ Harry J. Smith, Jul 20 2009

a(n) = floor(sqrt(prime(n+1))) - floor(sqrt(prime(n))). - Vladeta Jovovic, May 01 2003

Extended by Patrick De Geest, Jun 05 2001

Offset changed from 0 to 1 by Harry J. Smith, Jul 20 2009

A111208 Number of primes <= n-th triangular number.

Original entry on oeis.org

0, 0, 2, 3, 4, 6, 8, 9, 11, 14, 16, 18, 21, 24, 27, 30, 32, 36, 39, 42, 46, 50, 54, 58, 62, 66, 70, 74, 79, 84, 90, 94, 99, 102, 108, 114, 121, 126, 131, 137, 141, 149, 154, 160, 166, 174, 180, 188, 193, 200, 205, 216, 220, 226, 235, 242, 250, 259, 267, 274, 281, 290

Offset: 0

Giovanni Teofilatto, Oct 25 2005

nonn

Only because of the case n = 2 is it necessary to say "<=", otherwise "<" would suffice. Except for the first two terms, there are no consecutive identical terms for n < 10000. A065382 gives differences between consecutive terms of this sequence. - Alonso del Arte, Oct 31 2005

Harry J. Smith, Table of n, a(n) for n = 0..10000

Cf. A000720, A000217.

Cf. A038107, A014085, A194189.

a111208 n = length $ takeWhile (<= a000217 n) a000040_list
-- Reinhard Zumkeller, Nov 01 2011

Table[PrimePi[n*(n + 1)/2], {n, 0, 60}] (* Ray Chandler, Oct 31 2005 *)

{ allocatemem(932245000); default(primelimit, 4294965247); write("b111208.txt", 0, " ", 0); for (n = 1, 10000, t=n*(n + 1)/2; a=primepi(t); write("b111208.txt", n, " ", a); ) } \\ Harry J. Smith, Mar 10 2009

[prime_pi(binomial(n,2)) for n in range(1, 63)] # Zerinvary Lajos, Jun 06 2009

a(n) = A000720(A000217(n)).

Extended by Ray Chandler and Alonso del Arte, Oct 31 2005

A077766 Number of primes of form 4k+1 between n^2 and (n+1)^2.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 2, 3, 1, 2, 3, 1, 3, 4, 3, 3, 3, 4, 3, 2, 3, 5, 4, 3, 5, 4, 4, 4, 5, 4, 6, 5, 5, 4, 5, 4, 3, 7, 7, 3, 7, 5, 6, 5, 8, 8, 5, 4, 8, 9, 6, 5, 7, 7, 6, 8, 7, 8, 7, 6, 8, 7, 9, 8, 7, 7, 8, 9, 5, 10, 8, 7, 11, 9, 6, 10, 12, 8, 10, 10, 7, 8, 10, 12, 10, 11, 11, 9, 10, 10, 11, 10, 11, 11

Offset: 1

T. D. Noe, Nov 20 2002

nonn

Related to Legendre's conjecture that there is always a prime between two consecutive squares.

			a(8)=1 because the prime 73 is between squares 64 and 81.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A002144, A014085, A077767.

maxN=100; a=Table[0, {maxN}]; maxP=PrimePi[(maxN+1)^2]; For[i=1, i<=maxP, i++, p=Prime[i]; If[Mod[p, 4]==1, j=Floor[Sqrt[p]]; a[[j]]++ ]]; a

A094189 Number of primes between n^2-n and n^2 (inclusive).

Original entry on oeis.org

0, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 3, 2, 2, 2, 3, 4, 4, 3, 4, 3, 3, 4, 5, 4, 3, 4, 5, 4, 4, 5, 4, 4, 5, 5, 2, 6, 6, 5, 4, 6, 4, 5, 7, 7, 3, 7, 8, 4, 5, 10, 7, 5, 6, 5, 5, 10, 7, 8, 8, 6, 10, 7, 5, 5, 8, 7, 7, 5, 10, 7, 8, 10, 7, 7, 10, 10, 9, 12, 7, 11, 10, 10, 9, 7, 13, 11, 10, 10, 11, 10, 11, 10, 11

Offset: 1

Jason Earls, May 25 2004

Conjecture: for n>11, a(n)>1.

Oppermann conjectured in 1882 that a(n)>0 for n>1. - T. D. Noe, Sep 16 2008

Paulo Ribenboim, The New Book of Prime Number Records, 3rd ed., 1995, Springer, p. 248.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 183.

T. D. Noe, Table of n, a(n) for n = 1..10000
Wikipedia, Oppermann's conjecture

Cf. A014085, A089610, A108309, A010051.

a094189 n = sum $ map a010051' [n*(n-1) .. n^2]
-- Reinhard Zumkeller, Jun 07 2015

Table[PrimePi[n^2]-PrimePi[n^2-n-1],{n,100}] (* Harvey P. Dale, Jul 24 2015 *)

PARI
```
a(n) = sum(k=n^2-n,n^2,isprime(k))
```

a(n)=my(s);forprime(p=n^2-n,n^2,s++);s \\ Charles R Greathouse IV, Jan 18 2016

cp's OEIS Frontend

A089610 Number of primes between n^2 and (n+1/2)^2.

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A143223 (Number of primes between n^2 and (n+1)^2) - (number of primes between n and 2n).

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A143224 Numbers n such that (number of primes between n^2 and (n+1)^2) = (number of primes between n and 2n).

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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.

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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.

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A143935 Number of primes between n^K and (n+1)^K, inclusive, where K=log(127)/log(16).

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A061265 Number of squares between n-th prime and (n+1)st prime.

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