A144953 -id:A144953 - cp's OEIS frontend

A067200 Numbers m such that m^3 + 2 is prime.

0, 1, 3, 5, 29, 45, 63, 65, 69, 71, 83, 105, 113, 123, 129, 143, 153, 171, 173, 189, 209, 215, 219, 231, 243, 245, 249, 263, 291, 299, 305, 311, 341, 363, 369, 395, 419, 425, 431, 435, 473, 483, 491, 495, 501, 503, 509, 515, 533, 549, 555, 561, 575, 579, 639

Offset: 1

Benoit Cloitre, Feb 19 2002

nonn

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A144953.

Other sequences of the type "Numbers m such that m^k + k - 1 is prime": A000040 (k=1), A005574 (k=2), this sequence (k=3), A125259 (k=4), A125260 (k=5), A125261 (k=6), A125262 (k=7), A125263 (k=8), A125264 (k=10), A125265 (k=11).

lst={}; Do[If[PrimeQ[n^3+2], AppendTo[lst, n]], {n, 0, 10^3}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 08 2008 *)
Select[Range[0,700],PrimeQ[#^3+2]&] (* Harvey P. Dale, Jul 27 2025 *)

is(n)=isprime(n^3 + 2) \\ Charles R Greathouse IV, Apr 29 2015

a(n) = (A144953(n) - 2)^(1/3). - Zak Seidov, Sep 16 2013

A178251 Primes p such that p^3 - 2 is prime.

Original entry on oeis.org

19, 31, 37, 67, 109, 151, 211, 241, 277, 367, 439, 457, 619, 691, 727, 787, 859, 967, 1087, 1171, 1471, 1489, 1531, 1579, 1951, 2131, 2287, 2791, 2851, 2971, 3061, 3319, 3511, 3547, 3559, 3739, 4129, 4357, 4447, 4507, 4591, 4651, 4789, 4801, 4831, 4951

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 24 2010

			6857 = prime(882) = 19^3 - 2, 19 = prime(8) is 1st term.
29789 = prime(3228) = 31^3 - 2, 31 = prime(11) is 2nd term.

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

Cf. A000040, A000578, A038599, A038600, A144953.

[p: p in PrimesUpTo(5000) | IsPrime(p^3-2)]; // Vincenzo Librandi, Nov 17 2010

Select[Prime[Range[10000]], PrimeQ[#^3 - 2] &] (* Vincenzo Librandi, Mar 20 2014 *)

list(lim)=my(v=List()); forprime(p=2,lim, if(isprime(p^3-2), listput(v, p))); Vec(v) \\ Charles R Greathouse IV, Feb 08 2016

a = list(p for p in primes(10000) if is_prime(p**3-2)) # D. S. McNeil, May 25 2010

Base tag removed by D. S. McNeil, May 25 2010

A178336 Smaller member of a twin prime pair of the form (k^3 + 2, k^3 + 4).

Original entry on oeis.org

3, 29, 91127, 250049, 328511, 2146691, 47832149, 121287377, 170953877, 194104541, 693154127, 979146659, 1167575879, 1664006627, 5079577961, 6219352721, 8678316377, 10289109377, 10633486601, 13980103931, 17474794877, 28066748321, 28736971049

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 25 2010

nonn

			3 = 1^3+2 = prime(2) and 5 = 1^3+4 = prime(3) are a twin prime pair, so 3 becomes the first term.
91127 = 45^3+2 = prime(8811) and 91129 = 45^3+4 = prime(8812) are a twin prime pair, so 91127 is a term.

Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Band I, B. G. Teubner, Leipzig u. Berlin, 1909

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A013159, A053703, A132282, A144953, A173255, A178228.

Select[Range[3100]^3+2,PrimeQ[#]&&PrimeQ[#+2]&] (* Harvey P. Dale, May 26 2012 *)

a(n) = A178337(n)^3 + 2.

Keyword:base removed, 2 missing terms inserted by R. J. Mathar, Jun 27 2010

A178337 Numbers k such that (k^3 + 2, n^3 + 4) is a twin prime pair.

Original entry on oeis.org

1, 3, 45, 63, 69, 129, 363, 495, 555, 579, 885, 993, 1053, 1185, 1719, 1839, 2055, 2175, 2199, 2409, 2595, 3039, 3063, 3303, 3399, 3555, 3615, 4245, 4443, 4449, 5073, 5373, 5535, 5703, 5949, 6015, 6075, 6693, 6795, 6849, 7023, 7119, 7155, 7509, 7779, 8535

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 25 2010

nonn

With the exception of k = 1, all k are odd multiples of 3 with a least-significant decimal digit of 3, 5 or 9.

A178336(n) gives the values of k^3 + 2.

			1^3 + 2 = 3 = prime(2) and 3+2 = prime(3) are twin primes, so n=1 is a term.
45^3 + 2 = 91127 = prime(8811) and 91127+2 = prime(8812) are twin primes, so 45 is a term.
10893^3 + 2 = 1292535591959 = prime(48144179941) is a lower twin prime, so 10893 is a term.

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

Cf. A013159, A053703, A132282, A144953, A173255, A178336.

[n: n in [0..9000] | IsPrime(n^3+2) and IsPrime(n^3+4)]; // Vincenzo Librandi, Nov 18 2010

seqQ[n_] := And @@ PrimeQ[n^3 + 3 + {-1, 1}]; Select[Range[8535], seqQ] (* Amiram Eldar, Jan 11 2020*)

Keyword:base removed by R. J. Mathar, Jun 27 2010

A188764 Primes p such that all prime factors of p-2 have exponent 3.

Original entry on oeis.org

3, 29, 127, 24391, 274627, 328511, 357913, 571789, 1157627, 1442899, 1860869, 2146691, 2924209, 5177719, 9129331, 9938377, 10503461, 12326393, 15438251, 18191449, 24642173, 26730901, 28372627, 30080233, 39651823

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 09 2011

nonn

A048636 is the subsequence of terms where there is only one prime divisor of p-2. - M. F. Hasler, Jan 13 2025

			30080233-2 = 311^3, 39651823-2 = 11^3*31^3, ...
3-2 = 1 has no prime factors, so is trivially a member.

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

Subsequence of A144953; A048636 is a subsequence.

Cf. A089195, A188717.

Prepend[Select[Table[Prime[n],{n,3000000}],Length[Union[Last/@FactorInteger[#-2]]]==1&&Union[Last/@FactorInteger[#-2]]=={3}&], 3]
Prepend[Select[Prime[Range[25*10^5]],Union[FactorInteger[#-2][[All,2]]]=={3}&], 3] (* Harvey P. Dale, Nov 22 2018 *)
seq[lim_] := Select[Select[Range[Floor[Surd[lim-2, 3]]], SquareFreeQ]^3 + 2, PrimeQ]; seq[4*10^7] (* Amiram Eldar, Jan 18 2025 *)

list(lim)=my(v=List()); forsquarefree(k=1,sqrtnint(lim\1-2,3), my(p=k[1]^3+2); if(isprime(p), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Jan 14 2025

a(n) >> n^3. - Charles R Greathouse IV, Jan 14 2025

a(1) = 3 inserted by Charles R Greathouse IV, Jan 14 2025

A216977 Primes of the form n^5+2.

Original entry on oeis.org

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 */

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 */

A164520 Primes p such that p-2 is the product of exactly 2 distinct cubes of primes.

Original entry on oeis.org

274627, 328511, 1860869, 2146691, 2924209, 9129331, 9938377, 10503461, 15438251, 24642173, 26730901, 28372627, 39651823, 61629877, 105823819, 125751503, 136590877, 151419439, 194104541, 426957779, 573856193

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 14 2009

nonn

			274627 - 2 = 5^3*13^3, 328511 - 2 = 3^3*23^3,..

Chai Wah Wu, Table of n, a(n) for n = 1..10124

Cf. A056899, A144953, A164517, A164518, A164519

f3[n_]:=FactorInteger[n][[1,2]]==3&&Length[FactorInteger[n]]==2&&FactorInteger[n][[2,2]]==3; lst={};Do[p=Prime[n];If[f3[p-2],AppendTo[lst,p]],{n,4,4*9!}];lst

forprime(p=3,1e9,if(ispower(p-2,3,&n)&&!issquare(n)&&bigomega(n)==2,print1(p",")))

Program by Charles R Greathouse IV, Oct 12 2009

A176722 Primes of the form k^3 + 13, k >= 0.

Original entry on oeis.org

13, 229, 1013, 1741, 39317, 64013, 74101, 157477, 438989, 551381, 830597, 1906637, 2000389, 4096013, 7077901, 9261013, 10941061, 15625013, 16003021, 21024589, 24897101, 27000013, 69934541, 74088013, 79507013, 93576677, 122023949

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 25 2010

nonn

Necessarily, k = 6 * j or k = 6 * j + 4.

Values of k corresponding to terms of the sequence: 0, 6, 10, 12, 34, 40, 42, 54, 76, 82, 94, 124, 126, 160, 192, 210, 222, 250, 252, 276, 292, 300, 412, 420, 430, 454, 496, 502, 570, 586, 612, 622, 640, 670, 684, 712, 720, 724, 726, 756, 784, 822, 826, 874, 882, 894, 934, 952, 964, 1006, 1056.

			0^3 + 13 = 13 = prime(6) = a(1);
6^3 + 13 = 229 = prime(50) = a(2);
300^3 + 13 = 27000013 = prime(1683067) = a(22).

H. Rademacher, Topics in Analytic Number Theory, Springer-Verlag Berlin, 1973.

Robert Israel, Table of n, a(n) for n = 1..10000
D. R. Heath-Brown and B. Z. Moroz, Primes Represented by Binary Cubic Forms, Proceedings of the London Math. Soc. vol. 84(2), pp. 257-288, 2002; see also.

Cf. A000040, A000578, A037396, A138375, A144953, A154648, A157772, A168147, A168219, A168327.

[a: n in [0..500]|IsPrime(a) where a is n^3+13] // Vincenzo Librandi, Dec 22 2010

select(isprime,[seq(seq((6*j+m)^3+13,m=[0,4]),j=0..1000)]); # Robert Israel, Jun 28 2018

Select[Range[0,1000]^3+13,PrimeQ]  (* Harvey P. Dale, Mar 12 2011 *)

A178506 Lesser of a "near cube" twin prime pair (k^3 - 4, k^3 - 2).

Original entry on oeis.org

3371, 8120597, 69426527, 108531329, 176558477, 1207949621, 2379270371, 3477265871, 3560550179, 4227952109, 8012005997, 12665630687, 13060888871, 15832158827, 15945922409, 18337088849, 20279414579, 22354272509, 30283802609, 60559558979, 70496180087, 98035951127

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 29 2010

nonn

p + 2 = k^3 - 2 is form of "near(est) cube" prime smaller than cube number k^3, as k^3 - 1 = (k-1) * (k^2 + k + 1), only prime for k=2.

			p = 3371 = prime(475) = 15^3 - 4, (p, p+2) is twin prime pair tp(90), 3371 is the first term.

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

Cf. A178336.

Cf. A001359, A053703, A132282, A144953, A173255, A178228, A038600.

Select[Range[10^4]^3 - 4, And @@ PrimeQ[# + {0, 2}] &] (* Amiram Eldar, Dec 25 2019 *)

a(13) corrected and more terms from Amiram Eldar, Dec 25 2019

cp's OEIS Frontend

A067200 Numbers m such that m^3 + 2 is prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A178251 Primes p such that p^3 - 2 is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Extensions

A178336 Smaller member of a twin prime pair of the form (k^3 + 2, k^3 + 4).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A178337 Numbers k such that (k^3 + 2, n^3 + 4) is a twin prime pair.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

A188764 Primes p such that all prime factors of p-2 have exponent 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A216977 Primes of the form n^5+2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A216979 Primes of the form n^6+2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma