A007591 -id:A007591 - cp's OEIS frontend

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A114271 Numbers k such that k^2 + 8 is prime.

3, 9, 15, 21, 33, 51, 57, 81, 87, 111, 117, 123, 129, 135, 141, 147, 153, 177, 189, 213, 219, 255, 279, 285, 315, 321, 327, 345, 351, 363, 399, 417, 465, 471, 477, 483, 495, 549, 579, 585, 627, 657, 663, 669, 723, 735, 741, 747, 759, 771, 783, 789, 807, 825

Offset: 1

Zak Seidov, Nov 19 2005

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Other sequences of the type "Numbers k such that k^2 + i is prime": A005574 (i=1), A067201 (i=2), A049422 (i=3), A007591 (i=4), A078402 (i=5), A114269 (i=6), A114270 (i=7), this sequence (i=8), A114272 (i=9), A114273 (i=10), A114274 (i=11), A114275 (i=12).

a={};Do[If[PrimeQ[n^2+8], AppendTo[a, n]], {n,0,10^3}]; a (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Select[Range[1,901,2],PrimeQ[#^2+8]&] (* Harvey P. Dale, Aug 27 2023 *)

is(n)=isprime(n^2+8) \\ Charles R Greathouse IV, Jan 21 2015

A113536 Numbers k such that k^2 + 13 is prime.

Original entry on oeis.org

0, 2, 4, 10, 12, 16, 18, 28, 40, 42, 44, 46, 60, 68, 72, 82, 84, 88, 94, 108, 110, 114, 116, 122, 126, 142, 144, 152, 158, 180, 192, 194, 198, 200, 220, 222, 264, 266, 268, 282, 284, 296, 298, 332, 336, 340, 354, 378, 380, 418, 420, 430, 434, 446, 464, 466, 486

Offset: 1

Zak Seidov, Jan 13 2006

nonn

			If n=40 then n^2 + 13 = 1613 (prime), so 40 is in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Other cases: A005574 k=1, A067201 k=2, A049422 k=3, A007591 k=4, A078402 k=5, A114269-A114275 k=6-12.

With[{k=13}, Select[Range[1000], PrimeQ[ #^2+k]&]]

is(n)=isprime(n^2+13) \\ Charles R Greathouse IV, Feb 17 2017

Edited by R. J. Mathar, Aug 07 2008

A086381 Numbers n such that p=n^2+2 and p+2 are primes.

Original entry on oeis.org

1, 3, 15, 33, 45, 57, 117, 147, 243, 255, 303, 375, 423, 447, 453, 477, 573, 753, 837, 897, 903, 1035, 1497, 1905, 2055, 2085, 2193, 2283, 2433, 2487, 2535, 2583, 2757, 2823, 2943, 2955, 3003, 3213, 3285, 3345, 3603, 3657, 3687, 4407, 4575, 4977, 5037, 5043, 5325, 5355, 5367, 5403, 5727

Offset: 1

Zak Seidov, Sep 07 2003

The twin primes are given by A253639 and A085554. Except for the initial term, all a(n)=3 (mod 6). - M. F. Hasler, Jan 16 2015

M. F. Hasler, Table of n, a(n) for n = 1..10000

Cf. A253639, A085554, A059100, A087475.

[n: n in [0..10000]|IsPrime(n^2+2) and IsPrime(n^2+4)] // Vincenzo Librandi, Dec 16 2010

is_A086381(x)=ispseudoprime(x^2+2)&&ispseudoprime(x^2+4)
forstep(x=1,9999,2,is_A086381(x)&&print1(x",")) \\ M. F. Hasler, Jan 16 2015

Intersection of A067201 and A007591. - M. F. Hasler, Jan 19 2015

More terms from Vincenzo Librandi, Dec 16 2010

A121982 Numbers k such that k^2 + 15 is prime.

Original entry on oeis.org

2, 4, 8, 14, 16, 22, 26, 32, 34, 38, 44, 46, 52, 64, 68, 76, 86, 88, 98, 104, 106, 124, 134, 158, 172, 178, 184, 196, 202, 206, 212, 236, 238, 242, 248, 256, 262, 272, 284, 296, 298, 304, 316, 322, 326, 328, 338, 356, 362, 364, 374, 386, 388, 394, 398, 452, 472

Offset: 1

Parthasarathy Nambi, Sep 09 2006

			If k=104 then k^2 + 15 = 10831 (prime).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A005574, A067201, A049422, A007591, A078402, A114269, A114271, A114272, A114273, A114274, A114275, A113536, A121250.

[n: n in [0..6000] | IsPrime(n^2 + 15)]; // Vincenzo Librandi, Nov 13 2010

Select[Range[200], PrimeQ[#1^2 + 15] &] (* G. C. Greubel, Sep 13 2017 *)

is(n)=isprime(n^2+15) \\ Charles R Greathouse IV, Feb 17 2017

A121250 Numbers n such that n^2 + 14 is prime.

Original entry on oeis.org

3, 15, 27, 33, 45, 75, 87, 93, 165, 183, 195, 207, 243, 285, 297, 303, 345, 363, 375, 405, 435, 453, 495, 513, 537, 573, 585, 615, 627, 633, 657, 663, 717, 813, 843, 975, 1053, 1065, 1083, 1095, 1125, 1137, 1167, 1203, 1287, 1317, 1335, 1353, 1413, 1437, 1455

Offset: 1

Parthasarathy Nambi, Sep 06 2006

nonn

			If n=183 then n^2 + 14 = 33503 (prime).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A005574, A067201, A049422, A007591, A078402, A114269, A114271, A114272, A114273, A114274, A114275, A113536.

[n: n in [0..6000] | IsPrime(n^2+14)] // Vincenzo Librandi, Sep 03 2010

Select[Range[1,1500,2],PrimeQ[#^2+14]&] (* Harvey P. Dale, Aug 20 2011 *)

is(n)=isprime(n^2+14) \\ Charles R Greathouse IV, Feb 20 2017

A242332 Numbers k such that k^2 + 4 is a semiprime.

Original entry on oeis.org

0, 9, 19, 21, 23, 25, 31, 41, 43, 51, 53, 55, 63, 69, 71, 75, 77, 79, 83, 91, 93, 105, 107, 109, 113, 119, 123, 129, 131, 133, 143, 145, 149, 151, 153, 157, 165, 171, 173, 175, 181, 185, 187, 191, 195, 197, 201, 209, 221, 223, 225, 227, 241, 249, 251, 257, 259

Offset: 1

Vincenzo Librandi, May 14 2014

nonn

The semiprimes of this form are: 4, 85, 365, 445, 533, 629, 965, 1685, 1853, 2605, 2813, 3029, 3973, 4765, 5045, 5629, 5933, 6245, ...

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

Cf. A005473, A007591, A085722, A242330, A242331, A242333.

IsSemiprime:=func; [n: n in [0..300] | IsSemiprime(s) where s is n^2+4];

Mathematica

Select[Range[0, 300], PrimeOmega[#^2 + 4] == 2 &]

A263722 Integers k > 0 such that k^2 + p^2 is composite for all primes p.

Original entry on oeis.org

9, 11, 19, 21, 23, 25, 29, 31, 39, 41, 43, 49, 51, 53, 55, 59, 61, 63, 69, 71, 75, 77, 79, 81, 83, 89, 91, 93, 99, 101, 105, 107, 109, 111, 113, 119, 121, 123, 127, 129, 131, 133, 139, 141, 143, 145, 149, 151, 153, 157, 159, 161, 165, 169, 171, 173, 175, 179, 181, 185, 187, 189, 191, 195, 197, 199

Offset: 1

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Author

Jonathan Sondow and Robert G. Wilson v, Oct 24 2015

Keywords

nonn

Comments

Conjecture: All terms are odd. An equivalent conjecture in A263977 is that if k > 0 is even, then k^2 + p^2 is prime for some prime p. We have checked this for all k <= 12*10^7.

An odd number k is a term if and only if k^2 + 2^2 is composite, since k^2 + p^2 is even and > 2 (hence composite) for all odd primes p. Thus if the Conjecture is true, then the sequence is simply odd k such that k^2 + 4 is composite. The sequence of odd k, such that k^2 + 4 is prime, is A007591.

The complementary sequence is A263977. Given k in it, the smallest prime p, such that k^2 + p^2 is prime, is in A263726. These numbers k^2 + p^2 form A185086, the Fouvry-Iwaniec primes.

Examples

9^2 + 2^2 = 85 = 5*17, 11^2 + 2^2 = 125 = 5^3, and 23^2 + 2^2 = 533 = 13*41 are composite, so 9, 11, and 23 are members. 1^2 + 2^2 = 5 and 2^2 + 3^3 = 13 are prime, so 1, 2, and 3 are not members.

Links

Stephan Baier and Liangyi Zhao, On Primes Represented by Quadratic Polynomials, arXiv:math/0703284 [math.NT], 2007-2008; Anatomy of Integers, CRM Proc. & Lecture Notes, Vol. 46, Amer. Math. Soc. 2008, pp. 169 - 166.

Étienne Fouvry and Henryk Iwaniec, Gaussian primes, Acta Arithmetica 79:3 (1997), pp. 249-287.

Crossrefs

Cf. A002313, A045637, A062324, A185086, A007591, A240130, A240131, A263466, A263726, A263977.

A246519 Primes p such that 4+p, 4+p^2, 4+p^3 and 4+p^5 are all prime.

Original entry on oeis.org

7, 5503, 21013, 301123, 303613, 420037, 469363, 679153, 771427, 991957, 999667, 1524763, 1707367, 2030653, 2333083, 2540563, 2552713, 2710933, 3009967, 3378103, 3441817, 3592213, 4419937, 4704613, 4840723, 5177797, 5691547, 6227587, 6275887, 6395677, 6595597, 6597163

Offset: 1

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Author

Zak Seidov, Aug 28 2014

Keywords

nonn

Comments

For even k > 2, 4 + n^k is prime only for n = 1.

From Derek Orr, Aug 28 2014 (edited by Danny Rorabaugh, Apr 19 2015): (Start)

4+p^4 is composite for all primes p. For p = 2, 4+p^4 = 20 is composite. To prove it for odd primes, consider S(n) = 4+(2*n+1)^4. S(n) == 0 (mod 5) unless n == 2 (mod 5). If n == 2 (mod 5), then 2*n+1 == 0 (mod 5), which is only prime for n = 2; this gives p = 5 and 4+5^4 = 629 is composite. For other odd primes p, 4+p^4 is greater than 5 and divisible by 5.

4+p^(4*m) is also composite for any prime p and integer m > 0. For each m, the proof is the same as above.

(End)

All terms are == {3,7} (mod 10). - Zak Seidov, Aug 29 2014

Examples

From _K. D. Bajpai_, Jan 20 2015: (Start) a(2) = 5503: 4 + 5503 = 5507; 4 + 5503^2 = 30283013; 4 + 5503^3 = 166647398531; 4 + 5503^5 = 5046584669419727747; all five are prime. (End)

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007591, A073573, A125260, A172367.

Primes p such that 4+p^7, 4+p^9 and 4+p^11 are also prime is A253937. - K. D. Bajpai, Jan 20 2015

The subsequence with 4+p^7 also prime is A246562. - Danny Rorabaugh, Apr 19 2015

Programs

Magma

[p: p in PrimesUpTo(2*10^7) | IsPrime(4+p) and IsPrime(4+p^2) and IsPrime(4+p^3) and IsPrime(4+p^5)]; // Vincenzo Librandi, Apr 19 2015

Mathematica

k=4; Select[Prime[Range[1,500000]], PrimeQ[k+#]&&PrimeQ[k+#^2] &&PrimeQ[k+#^3] &&PrimeQ[k+#^5]&] (*K. D. Bajpai, Jan 20 2015 *)

PARI

for(n=1, 6000000, if(isprime(n) && isprime(4+n) && isprime(4+n^2) && isprime(4+n^3) && isprime(4+n^5), print1(n, ", "))) \\ Colin Barker, Aug 28 2014

PARI

p=7; forprime(q=11, 1e8, if(q-p==4 && isprime(4+p^2) && isprime(4+p^3) && isprime(4+p^5), print1(p, ", ")); p=q) \\ Charles R Greathouse IV, Aug 28 2014

Python

from sympy import prime, isprime A246519_list = [p for p in (prime(n) for n in range(1,10**5)) if all([isprime(4+p**z) for z in (1,2,3,5)])] # Chai Wah Wu, Sep 08 2014

A263977 Integers k > 0 such that k^2 + p^2 is prime for some prime p.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 14, 15, 16, 17, 18, 20, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 37, 38, 40, 42, 44, 45, 46, 47, 48, 50, 52, 54, 56, 57, 58, 60, 62, 64, 65, 66, 67, 68, 70, 72, 73, 74, 76, 78, 80, 82, 84, 85, 86, 87, 88, 90, 92, 94, 95, 96, 97, 98, 100, 102, 103, 104, 106, 108, 110, 112, 114, 115, 116, 117, 118, 120, 122, 124, 125, 126

Offset: 1

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Author

Jonathan Sondow and Robert G. Wilson v, Oct 30 2015

Keywords

nonn

Comments

The smallest such prime p is in A263726.

Complement of A263722.

An odd number k is a member if and only if k^2 + 4 is prime; see A007591.

Conjecture: Every even number k is a member. (This is equivalent to the Conjecture in A263722.) We have checked this for all k <= 12*10^7.

Examples

1^2 + 2^2 = 5, and 2 and 5 are prime, so a(1) = 1. 9^2 + p^2 is composite for all primes p, so 9 is not a member.

Links

Stephan Baier and Liangyi Zhao, On Primes Represented by Quadratic Polynomials, Anatomy of Integers, CRM Proc. & Lecture Notes, Vol. 46, Amer. Math. Soc. 2008, pp. 169 - 166.

Étienne Fouvry and Henryk Iwaniec, Gaussian primes, Acta Arithmetica 79:3 (1997), pp. 249-287.

Crossrefs

Cf. A007591, A240130, A240131, A263466, A263722, A263726.

Programs

Mathematica

fQ[n_] := Block[{p = 2}, While[ !PrimeQ[n^2 + p^2] && p < 1500, p = NextPrime@ p]; If[p > 1500, 0, p]]; lst = {}; k = 1; While[k < 130, If[fQ@ k > 0, AppendTo[lst, k]]; k++]; lst

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

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Author

Giovanni Teofilatto, Sep 12 2004

Keywords

nonn
easy

Comments

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

Examples

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

Links

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

Crossrefs

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

Programs

Magma

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

Mathematica

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

PARI

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

Extensions

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

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A114271 Numbers k such that k^2 + 8 is prime.

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A113536 Numbers k such that k^2 + 13 is prime.

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A086381 Numbers n such that p=n^2+2 and p+2 are primes.

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Magma

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A121982 Numbers k such that k^2 + 15 is prime.

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Magma

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PARI

A121250 Numbers n such that n^2 + 14 is prime.

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Magma

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PARI

A242332 Numbers k such that k^2 + 4 is a semiprime.

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Magma

Mathematica

A263722 Integers k > 0 such that k^2 + p^2 is composite for all primes p.

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A246519 Primes p such that 4+p, 4+p^2, 4+p^3 and 4+p^5 are all prime.

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