A056899 -id:A056899 - cp's OEIS frontend

A216977 Primes of the form n^5+2.

2, 3, 59051, 161053, 759377, 14348909, 90224201, 345025253, 601692059, 12762815627, 73439775751, 183765996901, 296709280759, 503756397101, 576650390627, 657748550153, 1572763671877, 1751989905403, 1880287678127, 2389769101501, 3101364196877, 3201078401359

Offset: 1

Michel Lagneau, Sep 21 2012

Subsequence of A053788. [Bruno Berselli, Sep 21 2012]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A053788, A056899, A144953, A216976.

[a: n in [0..400] | IsPrime(a) where a is n^5+2]; // Vincenzo Librandi, Mar 15 2013

lst={}; Do[p=n^5+2; If[PrimeQ[p], AppendTo[lst, p]], {n, 6!}]; lst
Select[Table[n^5 + 2, {n, 0, 400}], PrimeQ] (* Vincenzo Librandi, Mar 15 2013 *)

v=select(n->isprime(n^5+2),vector(2000,n,n-1)); /* A216976 */
vector(#v, n, v[n]^5+2)
/* Joerg Arndt, Sep 21 2012 */

A228244 Primes of the form k^2 + 17.

Original entry on oeis.org

17, 53, 593, 3617, 4373, 6101, 8117, 11681, 20753, 26261, 30293, 34613, 54773, 63521, 86453, 90017, 101141, 108917, 112913, 116981, 138401, 156833, 176417, 191861, 207953, 213461, 219041, 248021, 278801, 352853, 404513, 419921, 427733, 451601, 518417, 562517

Offset: 1

Michel Marcus, Aug 18 2013

			17 = 0^2 + 17 is prime.
53 = 6^2 + 17 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A049423, A056899, A056905, A079138, A138362, A138375, A241847, A264790.

[m: n in [0..900] | IsPrime(m) where m is n^2+17]; // Bruno Berselli, Aug 18 2013

Select[Table[n^2 + 17, {n, 0, 900}], PrimeQ] (* Bruno Berselli, Aug 18 2013 *)

PARI
```
isp(n) = isprime(n) && issquare(n-17);
```

a(n) = A241847(A264790(n)). - Elmo R. Oliveira, Apr 21 2025

A235640 Primes p of the form n^2 + 1234567890 where 1234567890 is the first pandigital number with digits in order.

Original entry on oeis.org

1234567891, 1234568059, 1234569571, 1234574779, 1234576171, 1234579771, 1234592539, 1234595779, 1234609099, 1234625011, 1234625971, 1234634971, 1234647979, 1234669651, 1234692499, 1234743451, 1234753651, 1234769491, 1234780411, 1234900819, 1234948579

Offset: 1

K. D. Bajpai, Apr 20 2014

nonn

			1234567891 is a prime and appears in the sequence because 1234567891 = 1^2 + 1234567890.
1234568059 is a prime and appears in the sequence because 1234568059 = 13^2 + 1234567890.

K. D. Bajpai, Table of n, a(n) for n = 1..6694

Cf. A000040, A002496, A028871, A050289, A056899, A240587, A234812.

KD := proc() local a; a:=n^2+1234567890; if isprime(a) then RETURN (a); fi; end: seq(KD(), n=1..2000);

Select[Table[k^2+1234567890,{k,1,2000}],PrimeQ]
c=0; a=n^2+1234567890; Do[If[PrimeQ[a],c=c+1; Print[c," ",a]], {n,0,200000}]  (*b-file*)

A059843 a(n) is the smallest prime p such that p-n is a nonzero square.

Original entry on oeis.org

2, 3, 7, 5, 41, 7, 11, 17, 13, 11, 47, 13, 17, 23, 19, 17, 53, 19, 23, 29, 37, 23, 59, 73, 29, 107, 31, 29, 173, 31, 47, 41, 37, 43, 71, 37, 41, 47, 43, 41, 617, 43, 47, 53, 61, 47, 83, 73, 53, 59, 67, 53, 89, 79, 59, 137, 61, 59, 383, 61, 97, 71, 67, 73, 101, 67, 71, 149, 73

Offset: 1

Labos Elemer, Feb 26 2001

nonn

			For n = 17, let P = {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,...} be the set of primes, then P - 17 = {-15,...,-4,0,2,6,12,14,20,24,26,30,36,...}. The first positive square in P - 17 is 36 with p = 53, so a(17) = 53. The square arising here is usually 1.

Zak Seidov, Table of n, a(n) for n = 1..10000

These terms arise in A002496, A056899, A049423, A005473, A056905, A056909 as first or 2nd entries depending on offset.
Cf. A002496, A056899, A049423, A005473, A056905, A056909.
Cf. A056896 (where p-n can be 0).

SearchLimit := 100;
for n from 1 to 400 do
k := 0: c := true:
while(c and k < SearchLimit) do
    k := k + 1:
    c := not isprime(k^2+n):
end do:
if k = SearchLimit then error("Search limit reached!") fi;
a[n] := k^2 + n end do: seq(a[j], j=1..400);
# Edited and SearchLimit introduced by Peter Luschny, Feb 05 2019

spsq[n_]:=Module[{p=NextPrime[n]},While[!IntegerQ[Sqrt[p-n]],p= NextPrime[ p]];p]; Array[spsq,70] (* Harvey P. Dale, Nov 10 2017 *)

for(n=1, 100, for(k=1, 100, if(isprime(k^2+n), print1(k^2+n, ", "); break()))) \\ Jianing Song, Feb 04 2019

a(n) = forprime(p=n,, if ((p-n) && issquare(p-n), return (p))); \\ Michel Marcus, Feb 05 2019

a(n) = min{p : p - n = x^2 for some x > 0, p is prime}.

Does a(n) exist for all n? - Jianing Song, Feb 04 2019

A098062 Primes of the form n^2 + 4n + 8.

Original entry on oeis.org

13, 29, 53, 173, 229, 293, 733, 1093, 1229, 1373, 2029, 2213, 3253, 4229, 4493, 5333, 7229, 7573, 9029, 9413, 10613, 13229, 13693, 15629, 18229, 18773, 21613, 24029, 26573, 27893, 31333, 33493, 37253, 41213, 42853, 46229, 47093, 54293, 55229

Offset: 1

Giovanni Teofilatto, Sep 12 2004

Or, primes that are equal to the mean of 7 consecutive squares. - Zak Seidov, Apr 14 2007
Sum of 7 consecutive squares starting with m^2 is equal to 7*(13 + 6*m + m^2) and mean is (13 + 6*m + m^2)=(m+3)^2+4. Hence a(n)=A005473(n+1). Note that only nonnegative m's are considered. - Zak Seidov, Apr 14 2007
a(n)==1 (mod 4).
a(n)= A005473(n+1). - Zak Seidov, Apr 12 2007

			13 = (0^2 + ... + 6^2)/7, 29 = (2^2 + ... + 8^2)/7 = 29, 53 = (4^2 + ... + 10^2)/7 = 53.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A005473, A056899, A067201, A007591, A129389, A129412.

[a: n in [0..250] | IsPrime(a) where a is n^2 + 4*n + 8]; // Vincenzo Librandi, Jul 17 2012

Select[ Table[ n^2 + 4n + 8, {n, 240}], PrimeQ[ # ] &] (* Robert G. Wilson v, Sep 14 2004 *)

for(n=0,240,if(isprime(p=n^2+4*n+8),print1(p,","))) \\ Klaus Brockhaus

Edited, corrected and extended by Robert G. Wilson v and Klaus Brockhaus, Sep 14 2004

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar

A129388 Primes that are equal to the mean of 5 consecutive squares.

Original entry on oeis.org

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, 178931

Offset: 1

Zak Seidov, Apr 12 2007

The sum of 5 consecutive squares starting with k^2 is equal to 5*(6 + 4*k + k^2) and the mean is (6 + 4*k + k^2) = (k+2)^2 + 2. Hence a(n)= A056899(n+2).

			11 = (1^2 + ... + 5^2)/5;
83 = (7^2 + ... + 11^2)/5;
227 = (13^2 + ... + 17^2)/5.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A056899, A067201, A102305, A129389.

[a: n in [1..600] | IsPrime(a) where a is  n^2 + 2*n + 3 ]; // Vincenzo Librandi, Mar 22 2013

Select[Table[n^2 + 2 n + 3, {n, 1, 600}], PrimeQ] (* Vincenzo Librandi, Mar 22 2013 *)

A102305=[n^2+2*n+3 for n in range(1,1001)]
[n^2+2*n+3 for n in (1..600) if is_prime(A102305[n-1])] # G. C. Greubel, Feb 03 2024

A132281 Noncomposites in A067200. Noncomposites (0, 1) and primes p such that A084380(p) = p^3 + 2 is prime.

Original entry on oeis.org

0, 1, 3, 5, 29, 71, 83, 113, 173, 263, 311, 419, 431, 491, 503, 509, 683, 701, 761, 773, 839, 911, 953, 1031, 1091, 1103, 1151, 1193, 1259, 1283, 1373, 1451, 1523, 1583, 1601, 1733, 1823, 1889, 1931, 2099, 2153, 2213, 2273, 2339, 2351, 2441, 2531, 2543

Offset: 1

Jonathan Vos Post, Aug 16 2007

The corresponding near-cube primes are A132282. Analog of near-square primes. After a(1) = 0, all values must be odd. Numbers of the form n^2+2 for n=1, 2, ... are 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, ... (A059100). These are prime for indices n = 1, 3, 9, 15, 21, 33, 39, 45, 57, 81, 99, ... (A067201), corresponding to the near-square primes 3, 11, 83, 227, 443, 1091, 1523, 2027, ... (A056899). Helfgott proves with minor conditions that: "Let f be a cubic polynomial. Then there are infinitely many primes p such that f(p) is squarefree." Note that 47^3 + 2 = 103825 = 5^2 * 4153 and similarly 97^3 + 2 is divisible by 5^2, but otherwise an infinite number of p^3+2 are squarefree.

			a(1) = 0 because 0^3 + 2 = 2 is prime and 0 is noncomposite.
a(2) = 1 because 1^3 + 2 = 5 is prime and 1 is noncomposite.
a(3) = 3 because 3^3 + 2 = 29 is prime and 3 is prime.
a(4) = 5 because 5^3 + 2 = 127 is prime and 5 is prime.
a(5) = 29 because 29^3 + 2 = 24391 is prime.
45 is not in the sequence because, although 45^3 + 2 = 91127 is prime, 45 is not prime.
63 is not in the sequence because, although 63^3 + 2 = 250049 is prime, 63 is not prime.
65 is not in the sequence because, although 65^3 + 2 = 274627 is prime, 65 is not prime.
a(6) = 71 because 71^3 + 2 = 357913 is prime.
a(7) = 83 because 83^3 + 2 = 571789 is prime.
a(8) = 113 because 113^3 + 2 = 1442899 is prime.
123 is not in the sequence because, although 123^3 + 2 = 1860869 is prime, 123 is not prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Harald Andres Helfgott, Power-free values, repulsion between points, differing beliefs and the existence of error, arXiv:0706.1497 [math.NT], 2007.

Cf. A000040, A056899, A059100, A067200, A067201, A084380, A132282.

{p in A000040 such that A067200(p) = A084380(p) = p^3 + 2 is in A000040}.

Union of {0,1} and A048637. - R. J. Mathar, Oct 18 2007

More terms from R. J. Mathar, Oct 18 2007

A146327 Numbers k such that the continued fraction of (1 + sqrt(k))/2 has period 2.

Original entry on oeis.org

2, 3, 10, 11, 12, 15, 21, 26, 27, 30, 35, 45, 50, 51, 56, 63, 77, 82, 83, 84, 87, 90, 93, 99, 117, 122, 123, 132, 143, 165, 170, 171, 182, 195, 221, 226, 227, 228, 230, 231, 235, 237, 240, 245, 255, 285, 290, 291, 306, 323, 357, 362, 363, 380, 399, 437, 442, 443

Offset: 1

Artur Jasinski, Oct 30 2008

nonn

For primes in this sequence see A056899, primes of the form k^2 + 2.

			a(1) = 2 because continued fraction of (1 + sqrt(2))/2 = 1, 4, 1, 4, 1, 4, 1, ... has repeating part (1,4), period 2.

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

Cf. A000290, A078370, A146326-A146345, A146348-A146360.

A146326 := proc(n) if not issqr(n) then numtheory[cfrac]( (1+sqrt(n))/2, 'periodic','quotients') ; nops(%[2]) ; else 0 ; fi; end: isA146327 := proc(n) RETURN(A146326(n) = 2) ; end: for n from 2 to 450 do if isA146327(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Sep 06 2009

Select[Range[1000], 2 == Length[ContinuedFraction[(1 + Sqrt[#])/2][[2]]] &]

226, 227, 290, 291 added by R. J. Mathar, Sep 06 2009

A164519 Primes p such that p+2 is the square of a product of 3 distinct primes.

Original entry on oeis.org

53359, 74527, 81223, 127447, 159199, 184039, 189223, 314719, 354023, 370879, 378223, 416023, 439567, 511223, 804607, 974167, 1046527, 1092023, 1177223, 1238767, 1535119, 1600223, 1718719, 2059223, 2082247, 2140367, 2223079

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 14 2009

nonn

Primes of the form A162143(k)-2.

			53359 + 2 = 3^2*7^2*11^2. 74527 + 2 = 3^2*7^2*13^2.

Chai Wah Wu, Table of n, a(n) for n = 1..11612

Cf. A056899, A164517, A164518

f[n_]:=FactorInteger[n][[1,2]]==2&&Length[FactorInteger[n]]==3&&FactorInteger[n][[2, 2]]==2&&FactorInteger[n][[3,2]]==2; lst={};Do[p=Prime[n];If[f[p+2], AppendTo[lst,p]],{n,4,9!}];lst

Definition rephrased by R. J. Mathar, Oct 21 2009

A216979 Primes of the form n^6+2.

Original entry on oeis.org

2, 3, 3518743763, 17596287803, 282429536483, 54980371265627, 93385106978411, 110322650964683, 151939915084883, 1363532208525371, 1870004703089603, 3684302682180851, 5257948522194371, 15813440003753003, 22416464978706683, 33227552537453171, 80425212553252451

Offset: 1

Michel Lagneau, Sep 21 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A056899, A144953, A216975, A216977.

[a: n in [0..700] | IsPrime(a) where a is n^6 + 2 ]; // Vincenzo Librandi, Oct 12 2012

lst={}; Do[p=n^6+2; If[PrimeQ[p], AppendTo[lst, p]], {n, 6!}]; lst
Select[Table[n^6 + 2, {n, 0, 700}], PrimeQ] (* Vincenzo Librandi, Oct 12 2012 *)

v=select(n->isprime(n^6+2),vector(2000,n,n-1)); /* A216978 */
vector(#v, n, v[n]^6+2)
/* Joerg Arndt, Sep 21 2012 */

cp's OEIS Frontend

A216977 Primes of the form n^5+2.

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A228244 Primes of the form k^2 + 17.

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A235640 Primes p of the form n^2 + 1234567890 where 1234567890 is the first pandigital number with digits in order.

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A059843 a(n) is the smallest prime p such that p-n is a nonzero square.

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A098062 Primes of the form n^2 + 4n + 8.

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A129388 Primes that are equal to the mean of 5 consecutive squares.

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A132281 Noncomposites in A067200. Noncomposites (0, 1) and primes p such that A084380(p) = p^3 + 2 is prime.

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