A053000 -id:A053000 - cp's OEIS frontend

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

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)

A070316 Numbers n such that nextprime(n^2)-n^2 (n>=1) sets a new record.

Original entry on oeis.org

1, 3, 5, 11, 18, 23, 42, 63, 69, 102, 128, 143, 226, 254, 349, 370, 590, 757, 833, 873, 1110, 1165, 5018, 14908, 49495, 87598, 106359, 185179, 269697, 558269, 1070246, 1673629, 4292936, 43957056, 148793437, 982920306, 1569443693, 4556758439

Offset: 1

Donald S. McDonald, May 11 2002

nonn

			nextprime(63^2) - 63^2 = 3989 - 3969 = 20, giving the terms 63 in the present sequence and 20 in A070317.

Charles R Greathouse IV, Table of n, a(n) for n = 1..45, Dec 29 2007

Positions of record values in A053000 for n >= 1. Cf. A070317.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; d = 0; Do[m = n; a = NextPrim[n^2] - n^2; If[a > d, d = a; Print[n]], {n, 1, 10^8}]

A070316(n,show_all=0)={my(k=0,r=0);for(n=1,n,until(nextprime(k++^2)-k^2>r ,); r=nextprime(k^2)-k^2; show_all&print1(k","));k} \\ M. F. Hasler, Mar 23 2013

Edited by N. J. A. Sloane and Robert G. Wilson v, May 11 2002
More terms from Ralf Stephan, Oct 14 2002
Further terms from Charles R Greathouse IV, Jun 16 2007
Typo in a(38) corrected by M. F. Hasler, Mar 23 2013

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

A070317 Record values of nextprime(n^2)-n^2, cf. A070316.

Original entry on oeis.org

1, 2, 4, 6, 7, 12, 13, 20, 22, 23, 27, 28, 33, 37, 42, 43, 49, 52, 54, 58, 71, 108, 147, 163, 202, 225, 232, 270, 292, 328, 331, 388, 541, 613, 712, 773, 780, 868, 869, 964, 993, 1024, 1045, 1065, 1083

Offset: 1

Donald S. McDonald, May 11 2002

nonn

			nextprime(63^2) - 63^2 = 3989 - 3969 = 20, giving the terms 63 in A070316 and 20 in the present sequence.

Cf. A070316.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; d = 0; Do[m = n; a = NextPrim[n^2] - n^2; If[a > d, d = a; Print[n]], {n, 1, 10^8}]

a(n) = A053000(A070316(n)). - M. F. Hasler, May 05 2013

Edited by N. J. A. Sloane and Robert G. Wilson v, May 11 2002
More terms from Ralf Stephan, Oct 14 2002
More terms from Charles R Greathouse IV, Jun 16 2007, Aug 08 2007
More terms (using A070316) from M. F. Hasler, May 05 2013

A132657 a(n) is the product of the least prime > n^2 and the greatest prime < (n+1)^2.

Original entry on oeis.org

6, 35, 143, 391, 899, 1739, 3233, 5293, 8051, 11413, 17653, 24883, 33389, 43931, 56977, 72731, 92881, 118829, 145699, 176039, 212197, 254701, 308911, 357163, 424663, 492179, 566609, 660293, 756611, 864371, 987307, 1120697, 1257923

Offset: 1

Jonathan Vos Post, Nov 15 2007

			a(1) = 6 = 2*3 = (smallest prime in [1^2,2^2]) * (largest prime in [1^2,2^2]).
a(2) = 35 = 5*7 = (smallest prime in [2^2,3^2]) * (largest prime in [2^2,3^2]).

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

Cf. A000040, A001358, A007491, A014085, A053000, A053001, A053607, A077766, A077767, A132435.

seq(nextprime(n^2)*prevprime((n+1)^2,n=1..100); # Robert Israel, Jan 26 2020

Table[Prime[PrimePi[n^2] + 1]*Prime[PrimePi[(n + 1)^2]], {n, 1, 40}] (* Stefan Steinerberger, Nov 20 2007 *)
NextPrime[#[[1]]]NextPrime[#[[2]],-1]&/@Partition[Range[40]^2,2,1] (* Harvey P. Dale, Aug 27 2022 *)

for(n=1,33,print1(nextprime(n^2)*precprime((n+1)^2),", ")) \\ Hugo Pfoertner, Jan 26 2020

a(n) = A007491(n) * A053001(n+1).

More terms from Stefan Steinerberger, Nov 20 2007

A144832 Distance from nxtprm(n^2) to (n+1)^2 in A144831 is prime.

Original entry on oeis.org

2, 5, 7, 11, 17, 17, 23, 29, 31, 41, 47, 67, 71, 71, 89, 89, 97, 113, 109, 107, 127, 131, 137, 157, 167, 173, 173, 191, 197, 193, 197, 227, 233, 227, 251, 257, 271, 293, 271, 307, 313, 317, 331, 349, 353, 383, 383, 409, 419, 431, 449, 463, 467, 487, 503, 509

Offset: 1

Enoch Haga, Sep 22 2008

			a(2)=5 because 3^2=9 and 4^2=16. Nxtprm(3^2)=11 and 16-11=5, a prime.

A007491 A053000 A144831

Terms in this sequence are from A144831 iff the distance from nxtprm n^2 to (n+1)^2 is prime.

A362222 Slowest increasing sequence where a(n) + n^2 is a prime.

Original entry on oeis.org

1, 3, 4, 7, 12, 17, 18, 19, 20, 27, 28, 29, 30, 31, 32, 37, 42, 43, 48, 49, 50, 57, 58, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 93, 96, 97, 100, 103, 104, 105, 124, 133, 138, 147, 148, 153, 154, 163, 166, 171, 184, 193, 196, 197, 198, 205

Offset: 1

Angad Singh, Apr 11 2023

nonn

			a(2) = 3, since the smallest number greater than all the previous terms which gives a prime when added to 2^2 is 3.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A053000, A107819.

R:= 1: t:= 1:
for n from 2 to 100 do
  t:= nextprime(t+n^2)-n^2;
  R:= R,t
od:
R; # Robert Israel, Apr 11 2023

a[n_] := a[n] = Module[{k = a[n - 1] + 1}, While[! PrimeQ[n^2 + k], k++]; k]; a[0] = 0; Array[a, 100] (* Amiram Eldar, Apr 12 2023 *)

seq(n)={my(a=vector(n), p=0); for(n=1, #a, p++; while(!isprime(p+n^2), p++); a[n]=p); a} \\ Andrew Howroyd, Apr 11 2023

from sympy import nextprime
from itertools import count, islice
def agen(): # generator of terms
    an = 1
    for n in count(2):
        yield an
        an = nextprime(an + n**2) - n**2
print(list(islice(agen(), 62))) # Michael S. Branicky, Apr 16 2023

A187409 n^2 + nextprime(n^2).

Original entry on oeis.org

3, 9, 20, 33, 54, 73, 102, 131, 164, 201, 248, 293, 342, 393, 452, 513, 582, 655, 728, 801, 884, 971, 1070, 1153, 1256, 1353, 1462, 1571, 1694, 1807, 1928, 2055, 2180, 2319, 2454, 2593, 2742, 2891, 3044, 3201, 3374, 3541, 3710, 3885, 4052, 4245, 4422, 4613

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 09 2011

nonn

			1^2+2=3, 2^2+5=9, 3^2+11=20,..

Cf. A053000, A056927, A007491.

Table[n2=n^2; NextPrime[n2]+n2, {n,100}]
#+NextPrime[#]&/@(Range[100]^2) (* Harvey P. Dale, Sep 20 2022 *)

A276556 a(n) = smallest prime p such that (smallest prime > p^2) == p^2 + 4n^2, n>=1.

Original entry on oeis.org

5, 281, 461, 4937, 25367, 75997, 1193909, 3464389, 48591863, 23674667, 22486333, 1648510979, 12708853771, 25139472583, 53498475287

Offset: 1

Zak Seidov, Apr 18 2017

			5^2+4*1^2=29, 281^2+4*2^2=78977, 461^2 + 4*3^2=212557 (all prime).

Cf. A007491, A053000.

Table[p = 2; While[NextPrime[p^2] != p^2 + 4 n^2, p = NextPrime@ p]; p, {n, 8}] (* Michael De Vlieger, Apr 22 2017 *)

a(n) = {forprime(p=2, , if (nextprime(p^2+1) == p^2 + 4*n^2, return (p)););} \\ Michel Marcus, Apr 19 2017

a(13)-a(15) from Rémy Sigrist, Apr 28 2017

cp's OEIS Frontend

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

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A070316 Numbers n such that nextprime(n^2)-n^2 (n>=1) sets a new record.

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

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A070317 Record values of nextprime(n^2)-n^2, cf. A070316.

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A132657 a(n) is the product of the least prime > n^2 and the greatest prime < (n+1)^2.

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A144832 Distance from nxtprm(n^2) to (n+1)^2 in A144831 is prime.

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A362222 Slowest increasing sequence where a(n) + n^2 is a prime.

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A187409 n^2 + nextprime(n^2).

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