A061235 -id:A061235 - cp's OEIS frontend

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

0, 2, 2, 2, 3, 2, 4, 3, 4, 3, 5, 4, 5, 5, 4, 6, 7, 5, 6, 6, 7, 7, 7, 6, 9, 8, 7, 8, 9, 8, 8, 10, 9, 10, 9, 10, 9, 9, 12, 11, 12, 11, 9, 12, 11, 13, 10, 13, 15, 10, 11, 15, 16, 12, 13, 11, 12, 17, 13, 16, 16, 13, 17, 15, 14, 16, 15, 15, 17, 13, 21, 15, 15, 17, 17, 18, 22, 14, 18, 23, 13

Offset: 0

Jon Wild, Jul 14 1997

Suggested by Legendre's conjecture (still open) that for n > 0 there is always a prime between n^2 and (n+1)^2.
a(n) is the number of occurrences of n in A000006. - Philippe Deléham, Dec 17 2003
See the additional references and links mentioned in A143227. - Jonathan Sondow, Aug 03 2008
Legendre's conjecture may be written pi((n+1)^2) - pi(n^2) > 0 for all positive n, where pi(n) = A000720(n), [the prime counting function]. - Jonathan Vos Post, Jul 30 2008 [Comment corrected by Jonathan Sondow, Aug 15 2008]
Legendre's conjecture can be generalized as follows: for all integers n > 0 and all real numbers k > K, there is a prime in the range n^k to (n+1)^k. The constant K is conjectured to be log(127)/log(16). See A143935. - T. D. Noe, Sep 05 2008
For n > 0: number of occurrences of n^2 in A145445. - Reinhard Zumkeller, Jul 25 2014

			a(17) = 5 because between 17^2 and 18^2, i.e., 289 and 324, there are 5 primes (which are 293, 307, 311, 313, 317).

J. R. Goldman, The Queen of Mathematics, 1998, p. 82.

T. D. Noe, Table of n, a(n) for n = 0..10000
Joel E. Cohen, Conjectures about Primes and Cyclic Numbers, arXiv:2508.08335 [math.NT], 2025. See p. 8.
Pierre Dusart, The k-th prime is greater than k(ln k + ln ln k-1) for k>=2, Mathematics of Computation 68: (1999), 411-415.
Tsutomu Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate, arXiv:0807.3690 [math.GM], 2008.
Mehdi Hassani, Counting primes in the interval (n^2, (n+1)^2), arXiv:math/0607096 [math.NT], 2006.
Edmund Landau, Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion. Jahresbericht der Deutschen Mathematiker-Vereinigung (1912), Vol. 21, page 208-228.
Peter Munn, Logarithmic plot: number of primes between consecutive squares vs number of integers between the same squares
Michael Penn, Legendre's Conjecture is probably true, and here's why, YouTube video, 2023.
Hugo Pfoertner, One million terms in b-file format, 13.1 MB (2024).
Hugo Pfoertner, Plot of 10^6 sequence terms with conjectured scatter band, large pdf (13.6 MB) (2024).
Hugo Pfoertner, Lower limit of the scatter band represented as a step function.
Jonathan Sorenson and Jonathan Webster, An algorithm to verify Legendre’s conjecture up to 7*10^13, Research in Number Theory 11:4 (2025).
Eric Weisstein's World of Mathematics, Legendre's Conjecture
Wikipedia, Legendre's conjecture

First differences of A038107.
Cf. A000006, A053000, A053001, A007491, A077766, A077767, A108954, A000720, A060715, A104272, A143223, A143224, A143225, A143226, A143227.
Cf. A010051, A061265, A221056, A000290, A145445.
Counts of primes between consecutive higher powers: A060199, A061235, A062517.
Cf. A333846, A349996, A349997, A349998, A349999.

a014085 n = sum $ map a010051 [n^2..(n+1)^2]
-- Reinhard Zumkeller, Mar 18 2012

Table[PrimePi[(n + 1)^2] - PrimePi[n^2], {n, 0, 80}] (* Lei Zhou, Dec 01 2005 *)
Differences[PrimePi[Range[0,90]^2]] (* Harvey P. Dale, Nov 25 2015 *)

a(n)=primepi((n+1)^2)-primepi(n^2) \\ Charles R Greathouse IV, Jun 15 2011

from sympy import primepi
def a(n): return primepi((n+1)**2) - primepi(n**2)
print([a(n) for n in range(81)]) # Michael S. Branicky, Jul 05 2021

a(n) = A000720((n+1)^2) - A000720(n^2). - Jonathan Vos Post, Jul 30 2008
a(n) = Sum_{k = n^2..(n+1)^2} A010051(k). - Reinhard Zumkeller, Mar 18 2012
Conjecture: for all n>1, abs(a(n)-(n/log(n))) < sqrt(n). - Alain Rocchelli, Sep 20 2023
Up to n=10^6 there are no counterexamples to this conjecture. - Hugo Pfoertner, Dec 16 2024
Sorenson & Webster show that a(n) > 0 for all 0 < n < 7.05 * 10^13. - Charles R Greathouse IV, Jan 31 2025

A060199 Number of primes between n^3 and (n+1)^3.

Original entry on oeis.org

0, 4, 5, 9, 12, 17, 21, 29, 32, 39, 49, 52, 58, 73, 76, 88, 92, 109, 117, 125, 140, 151, 159, 176, 188, 199, 207, 233, 247, 254, 267, 284, 305, 320, 346, 338, 373, 385, 416, 418, 437, 458, 481, 504, 517, 551, 555, 583, 599, 636, 648, 678, 686, 733, 723, 753, 810

Offset: 0

Labos Elemer, Mar 19 2001

nonn

Ingham showed that for n large enough and k=5/8, prime(n+1)-prime(n) = O(prime(n)^k). Ingham's result implies that there is a prime between sufficiently large consecutive cubes. Therefore a(n) is nonzero for n sufficiently large. Using the Riemann Hypothesis, Caldwell and Cheng prove there is a prime between all consecutive cubes. The question is undecided for squares. Many authors have reduced the value of k. The best value of k is 21/40, proved by Baker, Harman and Pintz in 2001. - corrected by Jonathan Sondow, May 19 2013
Conjecture: There are always more than 3 primes between two consecutive nonzero cubes. - Cino Hilliard, Jan 05 2003
Dudek (2014), correcting a claim of Cheng, shows that a(n) > 0 for n > exp(exp(33.217)) = 3.06144... * 10^115809481360808. - Charles R Greathouse IV, Jun 27 2014
Cully-Hugill shows the above for n > exp(exp(32.892)) = 6.92619... * 10^83675518094285. - Charles R Greathouse IV, Aug 02 2021
Mossinghoff, Trudgian, & Yang improve this to n > exp(exp(32.76)) = 3.62275 * 10^73328286790528. - Charles R Greathouse IV, Jul 31 2024

			n = 2: there are 5 primes between 8 and 27, 11,13,17,19,23.
n = 9, n+1 = 10: PrimePi(1000)-PrimePi(729) = 168-129 = a(9) = 39.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
R. C. Baker, G. Harman and J. Pintz, The difference between consecutive primes, II, Proc. London Math. Soc. (3) 83 (2001), no. 3, 532-562.
Chris K. Caldwell and Yuanyou Cheng, Determining Mills's Constant and a Note on Honaker's Problem, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.1.
Y.-Y. F.-R. Cheng, Explicit Estimate on Primes between consecutive cubes, Rocky Mountain Journal of Mathematics 40:1 (2010), pp. 117-153. arXiv:0810.2113 [math.NT], 2008-2013.
Michaela Cully-Hugill, Primes between consecutive powers, arXiv:2107.14468 [math.NT]
Adrian Dudek, An explicit result for primes between cubes arXiv:1401.4233 [math.NT], 2014.
Adrian Dudek, An explicit result for primes between cubes, Functiones et Approximatio Commentarii Mathematici Vol. 55, Issue 2 (Dec 2016), pp. 177-197. See also Explicit Estimates in the Theory of Prime Numbers, arXiv:1611.07251 [math.NT], 2016; PhD thesis, Australian National University, 2016.
A. E. Ingham, On the difference between consecutive primes, Quart. J. Math. Oxford 8 (1937), 255-266.
MacTutor, A. E. Ingham Biography
Michael J. Mossinghoff, Timothy S. Trudgian, and Andrew Yang, Explicit zero-free regions for the Riemann zeta-function, arXiv preprint (2022). arXiv:2212.06867 [math.NT]

First differences of A038098.

Cf. A000720, A014085, A014220, A061235, A062517.

[0] cat [#PrimesInInterval(n^3, (n+1)^3): n in [1..70]]; // Vincenzo Librandi, Feb 13 2016

PrimePi[(#+1)^3]-PrimePi[#^3]&/@Range[0,60] (* Harvey P. Dale, Feb 08 2013 *)
Last[#]-First[#]&/@Partition[PrimePi[Range[0,60]^3],2,1] (* Harvey P. Dale, Feb 02 2015 *)

cubespr(n)= for(x=0,n, ct=0; for(y=x^3,(x+1)^3, if(isprime(y), ct++; )); if(ct>=0,print1(ct, ", ")))  \\ Cino Hilliard, Jan 05 2003

from sympy import primepi
def a(n): return primepi((n+1)**3) - primepi(n**3)
print([a(n) for n in range(57)]) # Michael S. Branicky, Jun 22 2021

Table[PrimePi[(j+1)^3]-PrimePi[j^3], {j, 1, 100}]

Corrected and added more detail to the Ingham references. - T. D. Noe, Sep 23 2008
Combined two comments, correcting a bad error in the first comment. - T. D. Noe, Sep 27 2008
Edited by N. J. A. Sloane, Jan 17 2009 at the suggestion of R. J. Mathar

A062517 Number of primes between n^5 and (n+1)^5.

Original entry on oeis.org

0, 11, 42, 119, 273, 540, 954, 1573, 2456, 3624, 5181, 7177, 9666, 12797, 16514, 21098, 26454, 32836, 40134, 48760, 58508, 69714, 82277, 96723, 112702, 130639, 150488, 172617, 197039, 223915, 253318, 285540, 320450, 358839, 400159, 445011, 493504

Offset: 0

Labos Elemer, Jul 10 2001

nonn

			a(1) = 11 the number of primes between 1 = 1^5 and 32 = 2^5.

Amiram Eldar, Table of n, a(n) for n = 0..4000 (terms 0..83 from Harry J. Smith)

Cf. A014085, A060199, A061235.

Table[PrimePi[(w+1)^5]-PrimePi[w^5], {w, 0, 50}]

a(n) = { primepi((n + 1)^5) - primepi((n)^5) } \\ Harry J. Smith, Aug 08 2009

Edited for consistency by Peter Munn, Apr 30 2017

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

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A060199 Number of primes between n^3 and (n+1)^3.

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A062517 Number of primes between n^5 and (n+1)^5.

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