A007491 -id:A007491 - cp's OEIS frontend

A084597 Largest k such that there are exactly n primes between k^2 and (k+1)^2.

5, 9, 14, 17, 23, 26, 30, 42, 49, 55, 56, 80, 77, 72, 85, 84, 89, 119, 102, 118, 137, 136, 143, 140, 149, 156, 174, 178, 188, 184, 194, 200, 195, 207, 219, 198, 228, 247, 261, 263, 245, 249, 279, 297, 289, 327, 306, 310, 325, 335, 321, 290, 356, 344, 425, 365

Offset: 2

Harry J. Smith, May 31 2003

nonn

a(n) is the index of last occurrence of n in A014085. This sequence relies on a heuristic calculation and there is no proof that it is correct. Conjecture: There is no k that has only one prime between k^2 and (k+1)^2.

			a(14)=77 because 14 is in sequence A014085 for the last time at item 77. There are 14 primes between 77^2 and 78^2.

P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 143.

T. D. Noe, Table of n, a(n) for n = 2..1000
Eric Weisstein's World of Mathematics, Landau's Problems.

Cf. A007491, A014085, A076957, A084596.

A056928 Average of the smallest prime greater than n^2 and the largest prime less than n^2.

Original entry on oeis.org

4, 9, 15, 26, 34, 50, 64, 81, 99, 120, 144, 170, 195, 225, 254, 288, 324, 363, 399, 441, 483, 532, 574, 625, 675, 730, 780, 846, 897, 960, 1026, 1089, 1158, 1226, 1294, 1370, 1443, 1517, 1599, 1681, 1768, 1854, 1941, 2022, 2121, 2210, 2303, 2405, 2490

Offset: 2

Henry Bottomley, Jul 12 2000

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

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

Cf. A007491, A053000, A053001, A056927, A056929, A056930, A056931.

Table[n2=n^2;(NextPrime[n2,-1]+NextPrime[n2])/2,{n,2,100}] (* Vladimir Joseph Stephan Orlovsky, Mar 09 2011 *)

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

a(n) = (A007491(n) - A053001(n))/2.
a(n) = A000290(n) + (A053000(n) - A056927(n))/2.
a(n) = A000290(n) - A056929(n).

A056930 Average of smallest prime greater than n^2 and largest prime less than (n+1)^2.

Original entry on oeis.org

6, 12, 20, 30, 42, 57, 73, 90, 107, 133, 158, 183, 210, 239, 270, 305, 345, 382, 420, 461, 505, 556, 598, 652, 702, 753, 813, 870, 930, 994, 1059, 1122, 1193, 1260, 1332, 1406, 1479, 1560, 1635, 1726, 1812, 1897, 1983, 2070, 2168, 2255, 2354, 2444

Offset: 2

Henry Bottomley, Jul 12 2000

a(1)=2.5 which is not an integer

			a(4)=1 because smallest prime greater than 4^2 is 17, largest prime less than 5^2 is 23 and average of 17 and 23 is 20

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Cf. A002378, A007491, A053000, A053001, A056927, A056928, A056929, A056931.

Table[Mean[{NextPrime[n^2],NextPrime[(n+1)^2,-1]}],{n,2,50}] (* Harvey P. Dale, May 10 2019 *)

a(n) =(A007491(n)+A053001(n+1))/2 =A002378(n)-A056931(n)

A062772 Smallest prime larger than square of n-th prime.

Original entry on oeis.org

5, 11, 29, 53, 127, 173, 293, 367, 541, 853, 967, 1373, 1693, 1861, 2213, 2819, 3491, 3727, 4493, 5051, 5333, 6247, 6899, 7927, 9413, 10211, 10613, 11467, 11887, 12781, 16139, 17167, 18773, 19333, 22229, 22807, 24659, 26573, 27893, 29947, 32051

Offset: 1

Labos Elemer, Jul 18 2001

Subsequence of A007491. - Zak Seidov, Apr 30 2015

			100th prime, 541 immediately follows 529, square of 9th prime.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A054270, A054271, A000879.

Cf. A007491. - Zak Seidov, Apr 30 2015

with(numtheory): [seq(nextprime(ithprime(w)^2),w=1..100)];

Array[NextPrime[Prime[#]^2] &, 41] (* Michael De Vlieger, Nov 02 2017 *)

a(n) = { nextprime(prime(n)^2) } \\ Harry J. Smith, Aug 10 2009

a(n) = A007918(A001248(n)) = A151800(A001248(n)). - Michel Marcus, Jun 24 2014

a(n) = A007491(A000040(n)). - Zak Seidov, Apr 30 2015

A090119 a(n) = nextprime(A090117(n)), the smallest prime following squares listed in A090117 and also the distance of a(n) from the preceding prime is 2*n.

Original entry on oeis.org

5, 11, 29, 367, 149, 631, 127, 1949, 541, 907, 3251, 1693, 2503, 10427, 5779, 10831, 10007, 22229, 30631, 25301, 121123, 76207, 93047, 157627, 212557, 35729, 119027, 1121509, 190979, 672439, 693943, 1004027, 259099, 1646101, 675713, 1207841

Offset: 1

Labos Elemer, Jan 09 2004

nonn

			a(7) = 127 because 127-113 = 14 = 2*7 and 121 = 11^2 is between {127,113} closest primes to 121 a suitable square number. Also 127 is the smallest prime with this property.

Cf. A090116, A090117, A090118, A007917, A007918, A000720, A000040, A053001, A007491, A000290.

pre[x_] := Prime[PrimePi[x]]; nex[x_] := Prime[PrimePi[x]+1]; de[x_] := Prime[PrimePi[x]+1]-Prime[PrimePi[x]]; de[1] = 0; t=Table[de[w^2], {w, 1, 50000}]; mt=Table[Min[Flatten[Position[t, 2*j]]], {j, 1, 100}]; Table[nex[Part[mt, j]^2], {j, 1, Length[mt]}]

a(n) = nextprime(A090117(n)) = nextprime(A090116(n)^2).

a(n) = A007918(A090117(n)) = prime(1+pi(A090117(n))).

Name corrected by Jason Yuen, Jun 23 2025

A144831 (n+1)^2 - (smallest prime > n^2).

Original entry on oeis.org

2, 4, 5, 8, 7, 12, 11, 14, 17, 20, 17, 20, 23, 28, 29, 32, 31, 30, 33, 40, 41, 42, 35, 48, 45, 52, 51, 54, 47, 54, 57, 58, 65, 62, 67, 72, 71, 74, 77, 80, 71, 72, 75, 76, 89, 80, 91, 92, 89, 98, 95, 102, 97, 108, 99, 112, 113, 110, 109, 114, 117, 122, 107, 126, 127, 132, 131

Offset: 1

Enoch Haga, Sep 21 2008

Suggested by Conjecture 60 in Carlos Rivera's The Prime Puzzles & Problems Connection.

Legendre's conjecture that there is always a prime between n^2 and (n+1)^2 is equivalent to a(n) >= 0 for all n. As the conjecture is still opened, it is not proved that a(n) is nonn, although the keyword is automatically added. - Jean-Christophe Hervé, Oct 26 2013

			a(2)=4 because n=2, 2^2=4 and (2+1)^2=9. The gap in which primes are to be found is 4 - 9. Next prime=5 and 9-5=4. For a(3)=5, 3^2=9 and (3+1)^2=16. Next prime=11 and 16-11=5.

Jean-Christophe Hervé, Table of n, a(n) for n = 1..10000

Cf. A007491, A053000, A144832, A144256, A014085.

Table[n^2-NextPrime[(n-1)^2],{n,2,70}] (* Harvey P. Dale, Jan 22 2019 *)

a(n) = (n+1)^2 - nextprime(n^2); \\ Michel Marcus, Jun 08 2014

Calculate n^2 and (n+1)^2, e.g. 4 - 9. Find the next prime following n^2 and subtract from (n+1)^2. Next prime is 5 so 9-5=4, the distance from next prime to (n+1)^2.

a(n) = (n+1)^2 - A007491(n).

Definition rewritten by N. J. A. Sloane, Sep 28 2008

Definition rewritten by Jean-Christophe Hervé, Oct 26 2013

A053786 a(n) = next prime after n^4.

Original entry on oeis.org

2, 17, 83, 257, 631, 1297, 2411, 4099, 6563, 10007, 14653, 20743, 28571, 38431, 50627, 65537, 83537, 104987, 130337, 160001, 194483, 234259, 279847, 331777, 390647, 456979, 531457, 614657, 707293, 810013, 923539, 1048583, 1185929, 1336337, 1500643

Offset: 1

Enoch Haga, Mar 26 2000

Primes associated with A053785.

			a(5)=631 because 631 is the smallest prime larger than 5^4 = 625.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf.A000583, A007491, A007918.

[NextPrime(n^4): n in [1..35]]; // Bruno Berselli, Apr 26 2012

seq(nextprime(n^4),n=1..100); # Robert Israel, Jan 29 2018

NextPrime[Range[100]^4] (* Vladimir Joseph Stephan Orlovsky, Feb 25 2010 *)

vector(50, n, nextprime(n^4)) \\ Michel Marcus, Jan 09 2015

a(n) = A007918(A000583(n)). - Robert Israel, Jan 29 2018

Edited by Jon E. Schoenfield, Jan 09 2015

A053788 Next prime after n^5.

Original entry on oeis.org

2, 37, 251, 1031, 3137, 7789, 16811, 32771, 59051, 100003, 161053, 248839, 371299, 537841, 759377, 1048583, 1419877, 1889579, 2476121, 3200003, 4084109, 5153639, 6436351, 7962637, 9765629, 11881379, 14348909, 17210377, 20511157, 24300007

Offset: 1

Enoch Haga, Mar 26 2000

Primes associated with A053787.

			a(3)=251 because the value of n^5 immediately preceding is 243.

Cf. A000584, A007491, A053787.

NextPrime[Range[100]^5] (* Vladimir Joseph Stephan Orlovsky, Feb 25 2010 *)

New name from Michel Marcus, Jun 06 2014

A084596 a(n) is the number of times n is in sequence A014085; i.e., there are exactly a(n) cases where there are exactly n primes between m^2 and (m+1)^2 for m >= 0.

Original entry on oeis.org

1, 0, 4, 3, 4, 4, 4, 5, 4, 7, 5, 5, 5, 7, 2, 7, 5, 6, 4, 4, 5, 10, 6, 9, 7, 5, 2, 5, 6, 6, 10, 4, 5, 11, 5, 3, 8, 3, 8, 9, 7, 10, 5, 4, 6, 8, 8, 5, 6, 10, 8, 9, 4, 4, 6, 7, 8, 7, 5, 10, 9, 9, 6, 8, 7, 7, 7, 8, 6, 3, 5, 8, 4, 8, 14, 8, 7, 9, 10, 6, 9, 6, 7, 6, 6, 8, 10, 4, 8, 7, 6, 8, 5, 14, 6, 7, 11, 7, 10, 8

Offset: 0

Harry J. Smith, May 31 2003

nonn

This sequences uses a finite number of terms of A014085 to conjecture the behavior of all terms of A014085. The first 10000 terms of this sequence were computed using 120000 terms of A014085. Legendre's conjecture is equivalent to a(0) remaining 1 for all terms of A014085. [Comment reworded by T. D. Noe, Sep 04 2008]

			a(14)=2 because 14 is in sequence A014085 only two times. There are 14 primes between 64^2 and 65^2 as well as between 77^2 and 78^2. These are the only cases with exactly 14 primes.

P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 143.

T. D. Noe, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Landau's Problems.

Cf. A007491, A014085, A084597.

A090120 Numbers k such that nextprime(k^2) - prevprime(k^2) = 4.

Original entry on oeis.org

3, 4, 9, 10, 14, 15, 20, 21, 26, 33, 40, 110, 117, 124, 146, 206, 237, 250, 273, 303, 309, 326, 340, 350, 387, 429, 436, 440, 441, 447, 470, 513, 561, 573, 609, 634, 686, 704, 807, 897, 920, 1004, 1035, 1054, 1060, 1071, 1113, 1124, 1143, 1156, 1233, 1239

Offset: 1

Labos Elemer, Jan 09 2004

nonn

Note that the gap = 4 is partitioned either as 2+2 or as 3+1; 1+3 never occurs since n^2-1 is composite if n>2.

			k = 3 is a term since, k^2 = 9 is surrounded by the closest primes: {7,[9],11}.
k = 10 is a term since k^2 = 100 is surrounded by {97,[100],101}.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A090116, A090117, A090118, A090119, A007917, A007918, A000720, A000040, A053001, A007491, A000290.

Select[Range[3,1500], NextPrime[#^2] == NextPrime[#^2, -1] + 4 &] (* Giovanni Resta, May 26 2018 *)

isok(n) = nextprime(n^2) - precprime(n^2) == 4; \\ Michel Marcus, May 26 2018

Solutions to {x; A007918(x^2)-A007917(x^2) = 4}.

cp's OEIS Frontend

A084597 Largest k such that there are exactly n primes between k^2 and (k+1)^2.

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A056928 Average of the smallest prime greater than n^2 and the largest prime less than n^2.

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A056930 Average of smallest prime greater than n^2 and largest prime less than (n+1)^2.

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A062772 Smallest prime larger than square of n-th prime.

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A090119 a(n) = nextprime(A090117(n)), the smallest prime following squares listed in A090117 and also the distance of a(n) from the preceding prime is 2*n.

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A144831 (n+1)^2 - (smallest prime > n^2).

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A053786 a(n) = next prime after n^4.

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