A067201 -id:A067201 - cp's OEIS frontend

A132282 Near-cube primes: primes of the form p^3 + 2, where p is noncomposite.

2, 3, 29, 127, 24391, 357913, 571789, 1442899, 5177719, 18191449, 30080233, 73560061, 80062993, 118370773, 127263529, 131872231, 318611989, 344472103, 440711083, 461889919, 590589721, 756058033, 865523179, 1095912793

Offset: 1

Jonathan Vos Post, Aug 16 2007

The corresponding near-cube prime indices q are A132281. Analog of near-square primes. After a(1) = 2, 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) = 0^3 + 2 = 2 is prime and 0 is noncomposite.
a(2) = 1^3 + 2 = 3 is prime and 1 is noncomposite.
a(3) = 3^3 + 2 = 29 is prime and 3 is prime.
a(4) = 5^3 + 2 = 127 is prime and 5 is prime.
a(5) = 29^3 + 2 = 24391 is prime and 29 is prime.
45^3 + 2 = 91127 is prime, but not in this sequence because 45 is not prime.
63^3 + 2 = 250049 is prime, but not in this sequence because 63 is not prime.
a(6) = 71^3 + 2 = 357913 is prime.
a(7) = 83^3 + 2 = 571789 is prime.
a(8) = 113^3 + 2 = 1442899 is prime.

Harald Andres Helfgott, Power-free values, repulsion between points, differing beliefs and the existence of error, arXiv:0706.1497 [math.NT], 2007.

Cf. A000040, A048636, A056899, A059100, A067200, A067201, A084380, A132281.

Join[{2, 5}, Select[Prime[Range[200]]^3 + 2, PrimeQ[ # ] &]] (* Stefan Steinerberger, Aug 17 2007 *)

v=[2,3]; forprime(p=3, 1e4, if(isprime(t=p^3+2), v=concat(v, t))); t \\ Charles R Greathouse IV, Feb 14 2011

a(n) = A132281(n)^3 + 2. {p in A000040 such that for some q = 0, 1, or q in A000040, we have p = A067200(q) = A084380(q) = q^3 + 2 is in A000040}.

a(n) = A048636(n-2) for n >= 3. - Georg Fischer, Nov 03 2018

More terms from Stefan Steinerberger, Aug 17 2007

a(2) corrected by Charles R Greathouse IV, Feb 14 2011

A216974 Numbers k such that k^4+2 is prime.

Original entry on oeis.org

0, 1, 3, 9, 15, 21, 45, 57, 63, 69, 87, 99, 129, 141, 279, 285, 333, 345, 453, 459, 465, 471, 513, 519, 627, 657, 669, 693, 729, 771, 777, 783, 795, 801, 807, 873, 909, 921, 933, 969, 987, 1011, 1023, 1047, 1119, 1155, 1257, 1299, 1323, 1407, 1419, 1437, 1485

Offset: 1

Michel Lagneau, Sep 21 2012

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

Cf. A067200, A067201, A182343 (associated primes).

[n: n in [0..1500] | IsPrime(n^4+2)]; // Bruno Berselli, Sep 21 2012

lst={}; Do[If[PrimeQ[n^4+2], AppendTo[lst, n]], {n, 0, 10^3}]; lst
Select[Range[0, 1500], PrimeQ[#^4 + 2] &] (* Bruno Berselli, Sep 21 2012 *)

select(n->isprime(n^4+2),vector(2000,n,n-1)) /* Joerg Arndt, Sep 21 2012 */

A216978 Numbers n such that n^6+2 is prime.

Original entry on oeis.org

0, 1, 39, 51, 81, 195, 213, 219, 231, 333, 351, 393, 417, 501, 531, 567, 657, 729, 747, 807, 945, 1005, 1059, 1161, 1173, 1185, 1191, 1203, 1281, 1335, 1371, 1467, 1479, 1563, 1587, 1647, 1653, 1749, 1761, 1821, 1845, 1875, 1929, 2373, 2379, 2421, 2529, 2595

Offset: 1

Michel Lagneau, Sep 21 2012

nonn

Except for the first term, all terms must be odd numbers. - Harvey P. Dale, Sep 23 2012

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A067200, A067201, A216974, A216976.

lst={}; Do[If[PrimeQ[n^6+2], AppendTo[lst, n]], {n, 0, 3000}]; lst
Join[{0},Select[Range[1,3001,2],PrimeQ[#^6+2]&]] (* Harvey P. Dale, Sep 23 2012 *)

select(n->isprime(n^6+2),vector(2000,n,n-1)) /* Joerg Arndt, Sep 21 2012 */

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

A122062 Numbers k such that k^2 + 16 is prime.

Original entry on oeis.org

1, 5, 9, 11, 15, 21, 25, 29, 31, 41, 49, 51, 55, 65, 75, 79, 81, 89, 91, 95, 99, 109, 115, 119, 121, 125, 129, 151, 165, 179, 191, 211, 219, 221, 229, 231, 245, 249, 265, 275, 281, 289, 291, 295, 299, 301, 311, 315, 335, 351, 355, 361, 365, 369, 381, 389, 391

Offset: 1

Parthasarathy Nambi, Sep 14 2006

nonn

			If k=99 then k^2 + 16 = 9817 (prime).

Daniel Starodubtsev, Table of n, a(n) for n = 1..5000

Cf. A005574, A067201, A049422, A007591, A078402, A114269, A114271, A114272, A114273, A114274, A114275, A113536, A121250, A121982.

[n: n in [0..6000] | IsPrime(n^2 + 16)]; // Vincenzo Librandi, Nov 13 2010

Select[Range[1,501,2],PrimeQ[16+#^2]&] (* Harvey P. Dale, Dec 04 2018 *)

is(n)=isprime(n^2+16) \\ Charles R Greathouse IV, Jun 06 2017

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

A216980 Numbers n such that n^7+2 is prime.

Original entry on oeis.org

0, 1, 9, 21, 53, 63, 99, 123, 141, 155, 185, 213, 315, 363, 375, 449, 513, 521, 543, 555, 653, 669, 699, 731, 735, 759, 801, 843, 881, 975, 983, 995, 1031, 1095, 1115, 1131, 1149, 1161, 1221, 1253, 1395, 1413, 1451, 1473, 1491, 1571, 1599, 1625, 1659, 1733

Offset: 1

Michel Lagneau, Sep 21 2012

nonn

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

Cf. A067200, A067201, A216974, A216976, A216978.

lst={}; Do[If[PrimeQ[n^7+2], AppendTo[lst, n]], {n, 0, 10^3}]; lst
Select[Range[0,2000],PrimeQ[#^7+2]&] (* Harvey P. Dale, Mar 29 2016 *)

select(n->isprime(n^7+2),vector(2000,n,n-1)) /* Joerg Arndt, Sep 21 2012 */

A129412 Numbers k such that mean of 7 consecutive squares starting with k^2 is prime.

Original entry on oeis.org

0, 2, 4, 10, 12, 14, 24, 30, 32, 34, 42, 44, 54, 62, 64, 70, 82, 84, 92, 94, 100, 112, 114, 122, 132, 134, 144, 152, 160, 164, 174, 180, 190, 200, 204, 212, 214, 230, 232, 240, 242, 250, 252, 262, 264, 272, 274, 284, 290, 300, 304, 310, 314, 344, 354, 370, 372

Offset: 1

Zak Seidov, Apr 14 2007

Sum of 7 consecutive squares starting with k^2 is equal to 7*(13 + 6*k + k^2) and mean is (13 + 6*k + k^2) = (k+3)^2+4. Hence a(n) = A007591(n+1)-3.

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

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

Cf. A005473, A056899, A067201, A007591, A129389, A098062.

Select[Range[0,400],PrimeQ[Mean[Range[#,#+6]^2]]&]  (* Harvey P. Dale, Apr 23 2011 *)

is(n)=isprime(n^2+6*n+13) \\ Charles R Greathouse IV, Jun 13 2017

cp's OEIS Frontend

A129389 Numbers k such that the mean of 5 consecutive squares starting with k^2 is prime.

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A132282 Near-cube primes: primes of the form p^3 + 2, where p is noncomposite.

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A216974 Numbers k such that k^4+2 is prime.

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A216978 Numbers n such that n^6+2 is prime.

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

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A122062 Numbers k such that k^2 + 16 is prime.

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

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