A007491 -id:A007491 - cp's OEIS frontend

A113911 Prime numbers not appearing in the nextprime(x^2) sequence A007491.

3, 7, 13, 19, 23, 31, 41, 43, 47, 59, 61, 71, 73, 79, 89, 97, 103, 107, 109, 113, 131, 137, 139, 151, 157, 163, 167, 179, 181, 191, 193, 199, 211, 223, 229, 233, 239, 241, 251, 263, 269, 271, 277, 281, 283, 307, 311, 313, 317, 337, 347, 349, 353, 359, 373, 379

Offset: 1

Jorge Coveiro, Jan 29 2006

Complement[ Prime@Range@75, Prime[ PrimePi[ Range@19^2] + 1]] (* Robert G. Wilson v *)

is(n)=isprime(n)&&n>nextprime(sqrtint(n)^2) \\ Charles R Greathouse IV, Apr 09 2013

a(n) ~ n log n. - Charles R Greathouse IV, Apr 09 2013

More terms from Robert G. Wilson v, Jan 30 2006

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

Original entry on oeis.org

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

A056899 Primes of the form k^2 + 2.

Original entry on oeis.org

2, 3, 11, 83, 227, 443, 1091, 1523, 2027, 3251, 6563, 9803, 11027, 12323, 13691, 15131, 21611, 29243, 47963, 50627, 56171, 59051, 62003, 65027, 74531, 88211, 91811, 95483, 103043, 119027, 123203, 131771, 136163, 140627, 149771, 173891

Offset: 1

Henry Bottomley, Jul 05 2000

nonn

Also, primes of the form k^2 - 2k + 3.
Note that all terms after the first two are equal to 11 modulo 72 and that (a(n)-11)/72 is a triangular number, since they have to be 2 more than the square of an odd multiple of 3 to be prime, and if k = 6*m+3 then a(n) = k^2 + 2 = 72*m*(m+1)/2 + 11.
The quotient cycle length is 2 in the continued fraction expansion of sqrt(p) for these primes. E.g.: cfrac(sqrt(6563),6) = 81+1/(81+1/(162+1/(81+1/(162+1/(81+1/(162+`...`)))))). - Labos Elemer, Feb 22 2001
Primes in A059100; except for a(2)=3 a subsequence of A007491 and congruent to 2 modulo 9. For n>2, a(n)=11 (mod 72). - M. F. Hasler, Apr 05 2009

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988.
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano 1997.

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Near-Square Prime

Intersection of A146327 and A000040; intersection of A059100 and A000040.

Cf. A002496.

[n: n in PrimesUpTo(175000) | IsSquare(n-2)];  // Bruno Berselli, Apr 05 2011

[ a: n in [0..450] | IsPrime(a) where a is n^2 +2 ]; // Vincenzo Librandi, Apr 06 2011

select(isprime, [seq(t^2+2, t = 0..1000)]); # Robert Israel, Sep 03 2015

Select[ Range[0, 500]^2 + 2, PrimeQ] (* Robert G. Wilson v, Sep 03 2015 *)

print1("2, 3");forstep(n=3,1e4,6,if(isprime(t=n^2+2),print1(", "t))) \\ Charles R Greathouse IV, Jul 19 2011

For n>1, a(n) = 72*A000217(A056900(n-2))+11

a(n) = A067201(n)^2 + 2. - M. F. Hasler, Apr 05 2009

A053001 Largest prime < n^2.

Original entry on oeis.org

3, 7, 13, 23, 31, 47, 61, 79, 97, 113, 139, 167, 193, 223, 251, 283, 317, 359, 397, 439, 479, 523, 571, 619, 673, 727, 773, 839, 887, 953, 1021, 1087, 1153, 1223, 1291, 1367, 1439, 1511, 1597, 1669, 1759, 1847, 1933, 2017, 2113, 2207, 2297, 2399, 2477, 2593

Offset: 2

N. J. A. Sloane, Feb 21 2000

Suggested by Legendre's conjecture (still open) that there is always a prime between n^2 and (n+1)^2.

Legendre's conjecture is equivalent to a(n) > (n-1)^2. - John W. Nicholson, Dec 11 2013

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

T. D. Noe, Table of n, a(n) for n=2..1000

Cf. A007491, A053000, A014085.

Cf. A007917, A000290.

a053001 = a007917 . a000290  -- Reinhard Zumkeller, Jun 07 2015

Maple
```
[seq(prevprime(i^2),i=2..100)];
```

Table[Prime[PrimePi[n^2]], {n, 2, 60}] (* Stefan Steinerberger, Apr 01 2006 *)
Table[NextPrime[n^2, -1], {n, 2, 60}] (* Jean-François Alcover, Oct 14 2013 *)

a(n) = precprime(n^2) \\ Michel Marcus, Oct 14 2013

from sympy import prevprime
def a(n):  return prevprime(n*n)
print([a(n) for n in range(2, 52)]) # Michael S. Branicky, Jul 29 2022

a(n) = A007917(A000290(n)). - Reinhard Zumkeller, Jun 07 2015

More terms from James Sellers, Feb 22 2000

A053000 a(n) = (smallest prime > n^2) - n^2.

Original entry on oeis.org

2, 1, 1, 2, 1, 4, 1, 4, 3, 2, 1, 6, 5, 4, 1, 2, 1, 4, 7, 6, 1, 2, 3, 12, 1, 6, 1, 4, 3, 12, 7, 6, 7, 2, 7, 4, 1, 4, 3, 2, 1, 12, 13, 12, 13, 2, 13, 4, 5, 10, 3, 8, 3, 10, 1, 12, 1, 2, 7, 10, 7, 6, 3, 20, 3, 4, 1, 4, 13, 22, 3, 10, 5, 4, 1, 14, 3, 10, 5, 6, 21, 2, 9, 10, 1, 4, 15, 4, 9, 6, 1, 6, 3, 14

Offset: 0

N. J. A. Sloane, Feb 21 2000

Suggested by Legendre's conjecture (still open) that there is always a prime between n^2 and (n+1)^2.
Record values are listed in A070317, their indices in A070316. - M. F. Hasler, Mar 23 2013
Conjecture: a(n) <= 1+phi(n) = 1+A000010(n), for n>0. This improves on Oppermann's conjecture, which says a(n) < n. - Jianglin Luo, Sep 22 2023

J. R. Goldman, The Queen of Mathematics, 1998, p. 82.
R. K. Guy, Unsolved Problems in Number Theory, Section A1.

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A007491, A013632, A053001, A014085, A070316, A085099, A058055, A069003.

[NextPrime(n^2) - n^2: n in [0..100]]; // Vincenzo Librandi, Jul 06 2015

Maple
```
A053000 := n->nextprime(n^2)-n^2;
```

nxt[n_]:=Module[{n2=n^2},NextPrime[n2]-n2]
nxt/@Range[0,100]  (* Harvey P. Dale, Dec 20 2010 *)

A053000(n)=nextprime(n^2)-n^2  \\ M. F. Hasler, Mar 23 2013

from sympy import nextprime
def a(n): nn = n*n; return nextprime(nn) - nn
print([a(n) for n in range(94)]) # Michael S. Branicky, Feb 17 2022

a(n) = A013632(n^2). - Robert Israel, Jul 06 2015

More terms from James Sellers, Feb 22 2000

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

A056929 Difference between n^2 and average of smallest prime greater than n^2 and largest prime less than n^2.

Original entry on oeis.org

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

Offset: 2

Henry Bottomley, Jul 12 2000

Conjecture: the most frequent value will be 1 (including sequence variants with any even power n^2k). - Bill McEachen, Dec 12 2022

			a(4)=1 because smallest prime greater than 4^2 is 17, largest prime less than 4^2 is 13, average of 17 and 13 is 15 and 16-15=1.

Hugo Pfoertner, Table of n, a(n) for n = 2..10000

Cf. A007491, A053000, A053001, A056927, A056928, A056930, A056931.

with(numtheory): A056929 := n-> n^2-(prevprime(n^2)+nextprime(n^2))/2);

Array[# - Mean@ {NextPrime[#], NextPrime[#, -1]} &[#^2] &, 87, 2] (* Michael De Vlieger, May 20 2018 *)

a(n) = n^2 - (nextprime(n^2) + precprime(n^2))/2; \\ Michel Marcus, May 20 2018

a(n) = A000290(n) - A056928(n).

a(n) = (A056927(n) - A053000(n))/2.

More terms from James Sellers, Jul 13 2000

A065383 a(n) = smallest prime >= n*(n + 1)/2.

Original entry on oeis.org

2, 2, 3, 7, 11, 17, 23, 29, 37, 47, 59, 67, 79, 97, 107, 127, 137, 157, 173, 191, 211, 233, 257, 277, 307, 331, 353, 379, 409, 439, 467, 499, 541, 563, 599, 631, 673, 709, 743, 787, 821, 863, 907, 947, 991, 1039, 1087, 1129, 1181, 1229, 1277, 1327, 1381, 1433

Offset: 0

Reinhard Zumkeller, Nov 05 2001

nonn

Besides 7, terms exclude the greater of the twin primes (A006512). - Bill McEachen, Dec 01 2022

Harry J. Smith, Table of n, a(n) for n = 0..1000

See A097050 for another version.
Cf. A065382, A007491, A064384, A000217.
Cf. A000217.

a065383 n = head $ dropWhile (< a000217 n) a000040_list
-- Reinhard Zumkeller, Aug 03 2012

PrimeNext[n_]:=Module[{k=n},While[ !PrimeQ[k],k++ ];k];f[n_]:=n*(n+1)/2;lst={};Do[AppendTo[lst,PrimeNext[f[n]]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2010 *)
NextPrime/@(Accumulate[Range[0,60]]-1) (* Harvey P. Dale, Jul 31 2012 *)

{ for (n=0, 1000, write("b065383.txt", n, " ", nextprime(n*(n + 1)/2)) ) } \\ Harry J. Smith, Oct 17 2009

Edited by N. J. A. Sloane, Nov 21 2008

A056931 Difference between n-th oblong (promic) number, n(n+1), and the average of the smallest prime greater than n^2 and the largest prime less than (n+1)^2.

Original entry on oeis.org

0, 0, 0, 0, 0, -1, -1, 0, 3, -1, -2, -1, 0, 1, 2, 1, -3, -2, 0, 1, 1, -4, 2, -2, 0, 3, -1, 0, 0, -2, -3, 0, -3, 0, 0, 0, 3, 0, 5, -4, -6, -5, -3, 0, -6, 1, -2, 6, 2, -2, 1, -2, 0, 1, 9, 0, 2, -2, -3, 2, -1, -9, 1, 1, 2, -1, -6, -6, -1, -3, 0, 0, 0, 6, -1, -3, 3, -2, -7, 1, -2, 1, 2, -1, -4

Offset: 2

Henry Bottomley, Jul 12 2000

a(1)=-0.5 which is not an integer

			a(4)=0 because smallest prime greater than 4^2 is 17, largest prime less than 5^2 is 23, average of 17 and 23 is 20 and 4*5-20=0

Cf. A002378, A007491, A053000, A053001, A056927, A056928, A056929, A056930.

with(numtheory): A056931 := n-> n*(n+1)-(prevprime((n+1)^2)+nextprime(n^2))/2);

a(n) =A002378(n)-(A007491(n)+A053001(n+1))/2 =A002378(n)-A056930(n).

More terms from James Sellers, Jul 13 2000

A060272 Distance from n^2 to closest prime.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 6, 5, 2, 1, 2, 1, 4, 7, 2, 1, 2, 3, 6, 1, 6, 1, 2, 3, 2, 7, 6, 3, 2, 3, 2, 1, 2, 3, 2, 1, 12, 5, 2, 3, 2, 3, 2, 5, 2, 3, 8, 3, 6, 1, 2, 1, 2, 3, 10, 7, 2, 3, 2, 3, 4, 1, 4, 3, 2, 3, 2, 5, 4, 1, 2, 3, 2, 5, 6, 3, 2, 5, 6, 1, 4, 3, 4, 3, 2, 1, 6, 3, 2, 1, 4, 5, 4, 3, 2, 7, 8, 5, 2

Offset: 1

Labos Elemer, Mar 23 2001

nonn

			n=1: n^2=1 has next prime 2, so a(1)=1;
n=11: n^2=121 is between primes {113,127} and closer to 127, thus a(11)=6.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007491, A053001, A058043, A056927, A053000, A051700, A060268-A060269, A060272, A059959, A059790.

seq((s-> min(nextprime(s)-s, `if`(s>2, s-prevprime(s), [][])))(n^2), n=1..256);  # edited by Alois P. Heinz, Jul 16 2017

Table[Function[k, Min[k - #, NextPrime@ # - k] &@ If[n == 1, 0, Prime@ PrimePi@ k]][n^2], {n, 103}] (* Michael De Vlieger, Jul 15 2017 *)
Min[#-NextPrime[#,-1],NextPrime[#]-#]&/@(Range[110]^2) (* Harvey P. Dale, Jun 26 2021 *)

a(n) = if (n==1, nextprime(n^2) - n^2, min(n^2 - precprime(n^2), nextprime(n^2) - n^2)); \\ Michel Marcus, Jul 16 2017

a(n) = abs(A000290(n) - A113425(n)) = abs(A000290(n) - A113426(n)). - Reinhard Zumkeller, Oct 31 2005

cp's OEIS Frontend

A113911 Prime numbers not appearing in the nextprime(x^2) sequence A007491.

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

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A056899 Primes of the form k^2 + 2.

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A053001 Largest prime < n^2.

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A053000 a(n) = (smallest prime > n^2) - n^2.

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A014220 Next prime after n^3.

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