A067200 -id:A067200 - cp's OEIS frontend

A216976 Numbers k such that k^5+2 is prime.

0, 1, 9, 11, 15, 27, 39, 51, 57, 105, 149, 179, 197, 219, 225, 231, 275, 281, 285, 299, 315, 317, 321, 335, 369, 389, 401, 405, 411, 419, 425, 491, 509, 545, 561, 587, 725, 741, 779, 789, 819, 855, 879, 909, 915, 977, 1007, 1019, 1059, 1115, 1145, 1161, 1199

Offset: 1

Michel Lagneau, Sep 21 2012

nonn

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

Cf. A067200, A067201.

select(t->isprime(t^5+2), [$0..10000]); # Robert Israel, Jan 01 2021

lst={}; Do[If[PrimeQ[n^5+2], AppendTo[lst, n]], {n, 0, 10^3}]; lst
Select[Range[0,1200],PrimeQ[#^5+2]&] (* Harvey P. Dale, Jan 30 2025 *)

select(n->isprime(n^5+2),vector(2000,n,n-1)) /* Joerg Arndt, Sep 21 2012 */

A073573 Numbers n such that n^3 + 4 is prime.

Original entry on oeis.org

1, 3, 7, 9, 19, 25, 27, 45, 63, 69, 79, 85, 87, 99, 103, 109, 115, 117, 129, 133, 145, 169, 175, 183, 195, 207, 235, 277, 279, 283, 289, 295, 337, 343, 345, 355, 357, 363, 379, 469, 487, 495, 507, 519, 525, 529, 535, 537, 555, 559, 579, 645, 657, 663, 679, 703, 715, 717, 777, 783

Offset: 1

Zak Seidov, Sep 01 2002

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

Cf. A067200 (n^3+2 is prime).

[n: n in [0..1000]|IsPrime(n^3+4)]; // Vincenzo Librandi, Dec 16 2010

Select[ Range[ 550 ], PrimeQ[ #^3+4 ] & ]

is(n)=isprime(n^3+4) \\ Charles R Greathouse IV, Feb 20 2017

Corrected by Zak Seidov, Aug 31 2006

More terms from Vincenzo Librandi, Dec 16 2010

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

Original entry on oeis.org

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 */

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 */

A073574 Numbers n such that n^3 + 6 is prime.

Original entry on oeis.org

1, 5, 7, 13, 17, 41, 73, 77, 101, 103, 133, 143, 145, 161, 173, 181, 187, 251, 271, 283, 293, 313, 325, 391, 395, 425, 461, 497, 503, 511, 523, 581, 593, 595, 647, 671, 703, 733, 745, 803, 805, 815, 833, 847, 853, 875, 941, 965, 973, 997, 1001, 1021, 1085, 1091, 1097, 1111, 1141, 1183

Offset: 1

Zak Seidov, Sep 01 2002

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

Cf. A067200 (n^3+2 is prime).

[n: n in [0..1500]|IsPrime(n^3+6)]; // Vincenzo Librandi, Dec 16 2010

Select[ Range[ 950 ], PrimeQ[ #^3+6 ] & ]
Select[Range[1,1201,2],PrimeQ[#^3+6]&] (* Because all terms must be odd, there is no need to test even numbers *) (* Harvey P. Dale, Oct 03 2018 *)

is(n)=isprime(n^3+6) \\ Charles R Greathouse IV, Jun 12 2017

More terms from Vincenzo Librandi, Dec 16 2010

A269346 Perfect cubes that are not the difference of two primes.

Original entry on oeis.org

343, 729, 1331, 2197, 3375, 4913, 6859, 9261, 12167, 15625, 19683, 29791, 35937, 42875, 50653, 59319, 68921, 79507, 103823, 117649, 132651, 148877, 166375, 185193, 205379, 226981, 300763, 389017, 421875, 456533, 493039, 531441, 614125, 658503, 704969

Offset: 1

Waldemar Puszkarz, Feb 24 2016

nonn

An even number can be the difference of two primes, but an odd one can only be if an odd number m is such that m+2 is prime. Since a(n) is odd and such that a(n)+2 is composite, a(n) cannot be such a difference.
The cubes of this property are also the cubes in A269345.
It is still an open conjecture that every even number is the difference of 2 primes. On the other hand, a computer test shows that all even cubes <= 10^21 can be written as the difference of 2 primes. The computer program generating the sequence needs an additional part to test for even cubes besides checking that for odd m^3, m^3+2 is composite. - Chai Wah Wu, Mar 03 2016

			For n=1, 343 = 7^3 and 345 = 343+2 is a composite, so 343 is a term.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000578 (the cubes), A067200 (cube roots of terms that complement this sequence), A269345 (supersequence).

[n^3: n in [1..150 by 2] | not IsPrime(n^3+2)]; // Vincenzo Librandi, Feb 28 2016

Select[Range[1,125,2]^3, !PrimeQ[#+2]&]
Select[Range[125]^3, !PrimeQ[#+2]&&OddQ[#]&]
Select[Select[Range[2000000], OddQ[#]&& !PrimeQ[#]&& !PrimeQ[#+2]&], IntegerQ[CubeRoot[#]]&]

for(n=1, 125, n%2==1&&!isprime(n^3+2)&&print1(n^3, ", "))

A329727 Numbers k such that k^3 +- 2 and k +- 2 are prime.

Original entry on oeis.org

129, 1491, 1875, 2709, 5655, 6969, 10335, 14325, 14421, 17319, 26559, 35109, 37509, 43719, 50229, 52629, 101871, 102795, 104325, 105501, 120429, 127599, 132699, 136395, 137829, 157521, 172425, 173685, 179481, 186189, 191829, 211371, 219681, 221199, 229215, 234195

Offset: 1

K. D. Bajpai, Nov 19 2019

nonn

All terms in this sequence are divisible by 3.

			a(1) = 129:
  129^3 + 2 = 2146691;
  129^3 - 2 = 2146687;
  129   + 2 =     131;
  129   - 2 =     127; all four results are prime.
a(2) = 1491:
  1491^3 + 2 = 3314613773;
  1491^3 - 2 = 3314613769;
  1491   + 2 =       1493;
  1491   - 2 =       1489; all four results are prime.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Intersection of A038599, A067200, and A087679.

Cf. A040976, A052147, A090121, A268043, A268186.

[k:k in [1..250000]|forall{m:m in [-2,2]|IsPrime(k+m) and IsPrime(k^3+m)}]; // Marius A. Burtea, Nov 20 2019

Select[Range[500000], PrimeQ[#^3 + 2] && PrimeQ[#^3 - 2] && PrimeQ[# + 2] && PrimeQ[# - 2] &]

isok(k) = isprime(k-2) && isprime(k+2) && isprime(k^3-2) && isprime(k^3+2); \\ Michel Marcus, Nov 24 2019

list(lim)=my(v=List(),p=127,k); forprime(q=131,lim+2,if(q-p==4 && isprime((k=p+2)^3-2) && isprime(k^3+2), listput(v,k)); p=q); Vec(v) \\ Charles R Greathouse IV, May 06 2020

A073598 Numbers n such that n^3 + 5 is prime.

Original entry on oeis.org

2, 12, 14, 24, 26, 38, 42, 48, 56, 66, 78, 86, 92, 104, 116, 126, 138, 146, 164, 186, 192, 194, 198, 224, 242, 264, 276, 296, 324, 332, 386, 438, 488, 494, 498, 518, 524, 566, 576, 582, 588, 594, 596, 632, 684, 696, 698, 714, 716, 722, 728, 738, 758, 762, 806

Offset: 1

Zak Seidov, Sep 01 2002

For n^3+2 prime see A067200. For n^3+3 prime see A049441.

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

Cf. A067200, A049441.

[n: n in [1..900] | IsPrime(n^3 + 5)]; // Vincenzo Librandi, Sep 30 2012

Select[ Range[ 950 ], PrimeQ[ #^3+5 ] & ]

is(n)=isprime(n^3+5) \\ Charles R Greathouse IV, Jun 12 2017

cp's OEIS Frontend

A216976 Numbers k such that k^5+2 is prime.

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A073573 Numbers n such that n^3 + 4 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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A216980 Numbers n such that n^7+2 is prime.

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A073574 Numbers n such that n^3 + 6 is prime.

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A269346 Perfect cubes that are not the difference of two primes.

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