A060199 -id:A060199 - cp's OEIS frontend

A078507 Duplicate of A060199.

0, 4, 5, 9, 12, 17, 21, 29, 32, 39, 49, 52, 58, 73, 76, 88, 92, 109, 117, 125, 140, 151, 159

Offset: 0

dead

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

A068695 Smallest number (not beginning with 0) that yields a prime when placed on the right of n.

Original entry on oeis.org

1, 3, 1, 1, 3, 1, 1, 3, 7, 1, 3, 7, 1, 9, 1, 3, 3, 1, 1, 11, 1, 3, 3, 1, 1, 3, 1, 1, 3, 7, 1, 17, 1, 7, 3, 7, 3, 3, 7, 1, 9, 1, 1, 3, 7, 1, 9, 7, 1, 3, 13, 1, 23, 1, 7, 3, 1, 7, 3, 1, 3, 11, 1, 1, 3, 1, 3, 3, 1, 1, 9, 7, 3, 3, 1, 1, 3, 7, 7, 9, 1, 1, 9, 19, 3, 3, 7, 1, 23, 7, 1, 9, 7, 1, 3, 7, 1, 3, 1, 9, 3, 1

Offset: 1

Amarnath Murthy, Mar 03 2002

Max Alekseyev (see link) shows that a(n) always exists. Note that although his argument makes use of some potentially large constants (see the comments in A060199), the proof shows that a(n) exists for all n. - N. J. A. Sloane, Nov 13 2020

Many numbers become prime by appending a one-digit odd number. Some numbers (such as 20, 32, 51, etc.) require a 2-digit odd number (A032352 has these). In the first 100000 values of n there are only 22 that require a 3-digit odd number (A091089). There probably are some values that require odd numbers of 4 or more digits, but these are likely to be very large. - Chuck Seggelin, Dec 18 2003

			a(20)=11 because 11 is the minimum odd number which when appended to 20 forms a prime (201, 203, 205, 207, 209 are all nonprime, 2011 is prime).

David W. Wilson, Table of n, a(n) for n=1..10000
Max Alekseyev, Given n, there is a k such that the concatenation n||k is a prime, Nov 09 2020
Index entries for primes involving decimal expansion of n

Cf. A032352 (a(n) requires at least a 2 digit odd number), A091089 (a(n) requires at least a 3 digit odd number).

Cf. also A060199, A228325, A336893.

d[n_]:=IntegerDigits[n]; t={}; Do[k=1; While[!PrimeQ[FromDigits[Join[d[n],d[k]]]],k++]; AppendTo[t,k],{n,102}]; t (* Jayanta Basu, May 21 2013 *)
mon[n_]:=Module[{k=1},While[!PrimeQ[n*10^IntegerLength[k]+k],k+=2];k]; Array[mon,110] (* Harvey P. Dale, Aug 13 2018 *)

A068695=n->for(i=1,oo,ispseudoprime(eval(Str(n,i)))&&return(i)) \\ M. F. Hasler, Oct 29 2013

from sympy import isprime
from itertools import count
def a(n): return next(k for k in count(1) if isprime(int(str(n)+str(k))))
print([a(n) for n in range(1, 103)]) # Michael S. Branicky, Oct 18 2022

More terms from Chuck Seggelin, Dec 18 2003
Entry revised by N. J. A. Sloane, Feb 20 2006
More terms from David Wasserman, Feb 14 2006

A014220 Next prime after n^3.

Original entry on oeis.org

2, 2, 11, 29, 67, 127, 223, 347, 521, 733, 1009, 1361, 1733, 2203, 2749, 3389, 4099, 4919, 5839, 6863, 8009, 9277, 10651, 12197, 13829, 15629, 17579, 19687, 21961, 24391, 27011, 29803, 32771, 35951, 39313

Offset: 0

N. J. A. Sloane

nonn

According to Borwein's Remark 1, this is an example of a sequence of primes whose mean value is in [0,1]. - T. D. Noe, Sep 15 2008

More precisely, Borwein, Choi and Coons remark that the generalized Liouville function for this sequence has mean value in (0,1). - Jonathan Sondow, May 19 2013

Michael De Vlieger, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
Peter Borwein, Stephen K.K. Choi and Michael Coons, Completely multiplicative functions taking values in {-1,1}, arXiv:0809.1691 [math.NT], 2008.

Cf. A007491, A060199.

NextPrime[Range[0, 100]^3] (* Vladimir Joseph Stephan Orlovsky, Feb 25 2010 *)

a(n)=nextprime(n^3) \\ Charles R Greathouse IV, May 26 2015

a(n) < (n+1)^3 for n sufficiently large, by Ingham's theorem in A060199. - Jonathan Sondow, May 19 2013

A077037 Largest prime < n^3.

Original entry on oeis.org

7, 23, 61, 113, 211, 337, 509, 727, 997, 1327, 1723, 2179, 2741, 3373, 4093, 4909, 5827, 6857, 7993, 9257, 10639, 12163, 13807, 15619, 17573, 19681, 21943, 24379, 26993, 29789, 32749, 35933, 39301, 42863, 46649, 50651, 54869, 59281, 63997

Offset: 2

Reinhard Zumkeller, Oct 21 2002

nonn

Vincenzo Librandi, Table of n, a(n) for n = 2..1000

Cf. A053001, A014220, A000578, A060199.

PrimePrev[n_]:=Module[{k},k=n-1;While[ !PrimeQ[k],k-- ];k];f[n_]:=n^3;lst={};Do[AppendTo[lst,PrimePrev[f[n]]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 25 2010 *)
Table[NextPrime[n^3, -1], {n, 2, 40}] (* Robert G. Wilson v, Aug 17 2010 *)

a(n) = precprime(n^3); \\ Michel Marcus, Jan 14 2023

from sympy import prevprime
def a(n):  return prevprime(n**3)
print([a(n) for n in range(2, 41)]) # Michael S. Branicky, Jul 23 2021

a(n) > (n-1)^3 for all large n, by Ingham's theorem (see A060199). - Jonathan Sondow, Mar 27 2014

A038098 Number of primes < n^3.

Original entry on oeis.org

0, 4, 9, 18, 30, 47, 68, 97, 129, 168, 217, 269, 327, 400, 476, 564, 656, 765, 882, 1007, 1147, 1298, 1457, 1633, 1821, 2020, 2227, 2460, 2707, 2961, 3228, 3512, 3817, 4137, 4483, 4821, 5194, 5579, 5995, 6413, 6850, 7308, 7789, 8293

Offset: 1

Joe K. Crump (joecr(AT)carolina.rr.com)

nonn

From Zhi-Wei Sun, Oct 17 2015: (Start)
Conjecture: (i) For any integer k > 2 the sequence pi(n^k)/n^k (n = 2,3,...) is strictly decreasing, where pi(x) denotes the number of primes not exceeding x.
(ii) All the numbers pi(n^2)/n^2 (n = 1,2,3,...) are pairwise distinct. Moreover, we have pi(n^2)/n^2 > pi((n+1)^2)/(n+1)^2 for all n > 15646.
(End)

			a(2)=4 because the only primes < 8 are 2,3,5 and 7.

R. J. Mathar, Table of n, a(n) for n = 1..500

Cf. A014085, A038107, A060199 (first differences).

vector(100, n, primepi(n^3)) \\ Altug Alkan, Oct 17 2015

[prime_pi(n^3) for n in range(1, 45)] # Zerinvary Lajos, Jun 06 2009

a(n) = A000720(A000578(n)). - Michel Marcus, Sep 02 2013

A061235 Number of primes between n^4 and (n+1)^4.

Original entry on oeis.org

0, 6, 16, 32, 60, 96, 147, 207, 283, 382, 486, 619, 773, 945, 1139, 1351, 1610, 1870, 2165, 2496, 2848, 3237, 3653, 4125, 4572, 5118, 5698, 6269, 6894, 7586, 8309, 9033, 9907, 10656, 11616, 12522, 13509, 14552, 15639, 16708, 18009, 19140, 20527

Offset: 0

Amarnath Murthy, Apr 23 2001

nonn

			a(3) = 32, as the number of primes between 3^4 = 81 and 4^4 = 256 is 32.

Amiram Eldar, Table of n, a(n) for n = 0..10000 (terms 0..300 from Vincenzo Librandi)

Cf. A014085, A060199, A062517.

[0] cat [#PrimesInInterval(n^4, (n+1)^4): n in [1..50]]; // Vincenzo Librandi, Apr 30 2017

Table[PrimePi[(w+1)^4]-PrimePi[w^4], {w, 0, 100}]

a(n) = primepi((n+1)^4) - primepi(n^4); \\ Michel Marcus, Apr 29 2017

More terms from Labos Elemer, Jul 10 2001

Edited for consistency by Peter Munn, Apr 28 2017

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

A060303 Number of primes below n^2 does not exceed n times the number of primes below n.

Original entry on oeis.org

0, 0, 2, 2, 6, 7, 13, 14, 14, 15, 25, 26, 39, 40, 42, 42, 58, 60, 80, 82, 83, 84, 108, 111, 111, 112, 114, 115, 144, 146, 179, 180, 182, 183, 185, 186, 225, 228, 228, 229, 270, 272, 319, 321, 324, 325, 376, 378, 378, 383, 387, 387, 439, 443, 446, 451, 455, 454

Offset: 1

Labos Elemer, Mar 26 2001

nonn

			pi(100) = 25, 10*pi(10) = 40, a(10) = 40-25 = 15.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Sanford L. Segal, On Pi(x+y)<=Pi(x)+Pi(y), Transactions American Mathematical Society, Vol. 104, No. 3 (1962), pp. 523-527.

Cf. A000720, A060199, A060208.

Table[n*PrimePi[n]-PrimePi[n^2], {n, 1, 100}]

Offset corrected by Amiram Eldar, Sep 06 2024

A060304 Number of primes below n^3 does not exceed n times the number of primes below n^2.

Original entry on oeis.org

0, 0, 3, 6, 15, 19, 37, 47, 69, 82, 113, 139, 180, 216, 244, 300, 381, 423, 486, 553, 638, 726, 820, 887, 1029, 1152, 1256, 1376, 1527, 1659, 1794, 1992, 2156, 2357, 2517, 2739, 2909, 3085, 3365, 3627, 3933, 4200, 4380, 4687, 4960, 5313, 5547, 5917, 6395

Offset: 0

Labos Elemer, Mar 26 2001

nonn

			n=10, 10*pi(100)=250, pi(1000)=168, a(10)=250-168=82.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Segal, On π(x+y)<=π(x)+π(y), Transactions American Mathematical Society, 104 (1962), 523-527.

Cf. A060199, A060208, A000720.

Table[n*PrimePi[n^2]-PrimePi[n^3], {n, 1, 100}]

a(n) = n*pi(n*n) - pi(n*n*n). - Jonathan Sondow, Feb 17 2014

a(n) = n*A038107(n) - A038098(n). - Michel Marcus, Feb 17 2014

cp's OEIS Frontend

A078507 Duplicate of A060199.

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

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A068695 Smallest number (not beginning with 0) that yields a prime when placed on the right of n.

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A014220 Next prime after n^3.

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A077037 Largest prime < n^3.

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A038098 Number of primes < n^3.

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A061235 Number of primes between n^4 and (n+1)^4.

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