cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A057429 Numbers n such that (1+i)^n - 1 times its conjugate is prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 29, 47, 73, 79, 113, 151, 157, 163, 167, 239, 241, 283, 353, 367, 379, 457, 997, 1367, 3041, 10141, 14699, 27529, 49207, 77291, 85237, 106693, 160423, 203789, 364289, 991961, 1203793, 1667321, 3704053, 4792057, 15317227
Offset: 1

Views

Author

Robert G. Wilson v, Sep 07 2000

Keywords

Comments

Equivalently, numbers n such that (1+i)^n - 1 is a Gaussian prime.
Note that n must be a rational prime. Also note that (1+i)^n + i or (1+i)^n - i is also a Gaussian prime. - T. D. Noe, Jan 31 2005
Primes which are the norms of the Gaussian integers (1 + i)^n - 1 or (1 - i)^n - 1. - Jonathan Vos Post, Feb 05 2010
Let z = (1+i)^n - 1. The product of z and its conjugate is 1 + 2^n - cos(n*Pi/4)*2^(1+n/2). For n > 3, the primes are in A007670 or A007671 depending on whether n = {1, 7} (mod 8) or n = {3, 5} (mod 8), respectively. - T. D. Noe, Mar 07 2010
Primes p such that ((1+i)^p - 1)((1-i)^p - 1) is prime. Number 2 together with odd primes p such that the norm 2^p - (-1)^((p^2-1)/8)*2^((p+1)/2) + 1 is prime. Note that Legendre symbol (2/p) = (-1)^((p^2-1)/8) as above. - Thomas Ordowski, Feb 20 2013
The exhaustive search for all a(n)<5000000 is now complete. - Serge Batalov, Sep 06 2014
The primes generated by these series are also generalized unique primes. They can be represented as Phi(4, 2^((p+1)/2) - (2/p))/2, where (2/p) is the Legendre symbol (Cf. link to Generalized unique primes page at UTM). - Serge Batalov, Sep 08 2014

Examples

			Note that 4 is not in the sequence because (1+i)^4 - 1 = -5, which is an integer prime, but not a Gaussian prime.

References

  • Mike Oakes, posting to the Mersenne list, Sep 07 2000.

Links

Crossrefs

Cf. A000043, A066408, A007670, A007671, A027206.
Cf. A027206 ((1+i)^n + i is a Gaussian prime), A103329 ((1+i)^n - i is a Gaussian prime).

Programs

  • Mathematica
    Do[a = (1 + I)^n - 1; b = a * Conjugate[a]; If[PrimeQ[b], Print[n]], {n, 1, 160426}] (* Wilson *)
Select[Range[1000], PrimeQ[((1 + I)^# - 1)Conjugate[(1 + I)^# - 1]] &] (* Alonso del Arte, May 01 2014 *)
Select[Range[48*10^5],PrimeQ[(1+I)^#-1,GaussianIntegers->True]&] (* Harvey P. Dale, Dec 30 2018 *)
  • PARI
    N=10^7; default(primelimit,N);
forprime(p=2,N,if(ispseudoprime(norm((1+I)^p-1)),print1(p,", ")));
/* Joerg Arndt, Jul 06 2011 */

Extensions

364289 found by Nicholas Glover on Jun 02 2001 - Mike Oakes
Edited by Dean Hickerson, Aug 14 2002; revised by N. J. A. Sloane, Dec 28 2005
a(37)-a(38) from B. Jaworski (found in 2006 and 2011) - Serge Batalov, May 01 2014
a(39)-a(40) from Serge Batalov, Sep 06 2014
a(41) from Ryan Propper and Serge Batalov, Jun 20 2023

A125738 Primes p such that 3^p - 3^((p + 1)/2) + 1 is prime.

Original entry on oeis.org

3, 11, 193, 239, 659, 709, 1103, 2029, 9049, 10453, 255361, 534827, 2888387
Offset: 1

Views

Author

Alexander Adamchuk, Dec 02 2006

Keywords

Comments

PrimePi[ a(n) ] = {2, 5, 44, 52, 120, 127, 185, 308, 1125, 1278 ...}, the indices of the primes p.
a(14) > 4400000. - Serge Batalov, Jun 20 2023

Links

Crossrefs

Cf. A125739 = Primes p such that 3^p + 3^((p + 1)/2) + 1 is prime.
Cf. A007670 = Numbers n such that 2^n - 2^((n + 1)/2) + 1 is prime.
Cf. A007671 = Numbers n such that 2^n + 2^((n + 1)/2) + 1 is prime.
Cf. A066408 = Numbers n such that the Eisenstein integer has prime norm.

Programs

  • Mathematica
    Do[p=Prime[n];f=3^p-3^((p+1)/2)+1;If[PrimeQ[f],Print[{n,p}]],{n,1,200}]

Extensions

More terms from A066408 by Serge Batalov, Mar 24 2014
a(13) from Ryan Propper and Serge Batalov, Jun 20 2023

A125739 Primes p such that 3^p + 3^((p + 1)/2) + 1 is prime.

Original entry on oeis.org

3, 5, 7, 17, 19, 79, 163, 317, 353, 1049, 1759, 5153, 7541, 23743, 2237561, 4043119
Offset: 1

Views

Author

Alexander Adamchuk, Dec 02 2006

Keywords

Comments

PrimePi[ a(n) ] = {2, 3, 4, 7, 8, 22, 38, 66, 71, 176, 274, 687, 956, ...}, the indices of the primes p.
a(17) > 4400000. - Serge Batalov, Jun 20 2023

Links

Crossrefs

Cf. A125738 = Primes p such that 3^p - 3^((p + 1)/2) + 1 is prime.
Cf. A007670 = Numbers n such that 2^n - 2^((n + 1)/2) + 1 is prime.
Cf. A007671 = Numbers n such that 2^n + 2^((n + 1)/2) + 1 is prime.
Cf. A066408 = Numbers n such that the Eisenstein integer has prime norm.

Programs

  • Magma
    [p: p in PrimesUpTo(5000) | IsPrime(3^p+3^((p+1)div 2)+1)]; // Vincenzo Librandi, Oct 13 2014
  • Mathematica
    Do[p=Prime[n];f=3^p+3^((p+1)/2)+1;If[PrimeQ[f],Print[{n,p}]],{n,1,200}]
  • PARI
    lista(nn) = {forprime(p=3, nn, if (ispseudoprime(3^p + 3^((p + 1)/2) + 1), print1(p, ", ")););} \\ Michel Marcus, Oct 13 2014

Extensions

a(11)-a(13) from Stefan Steinerberger, Sep 08 2007
a(14) from Lelio R Paula (lelio(AT)sknet.com.br), May 07 2008
a(15) from Serge Batalov, Oct 12 2014
a(16) from Ryan Propper and Serge Batalov, Jun 20 2023

A006598 Numbers n such that 2^(2n+1) - 2^(n+1) + 1 is a prime.

Original entry on oeis.org

1, 3, 23, 36, 39, 56, 75, 83, 119, 120, 176, 183, 228, 683, 1520
Offset: 1

Views

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

Comments

These numbers satisfy A100014(n)=2. - Michel Marcus, Mar 07 2013

References

  • J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A007670.
Indices of primes in A092440. For the actual primes see A325914.

Programs

  • Mathematica
    Do[ If[ PrimeQ[ 2^(2n + 1) - 2^(n + 1) + 1 ], Print[n] ], {n, 1, 4000} ]

A125743 Primes p such that (3^p - 3^((p + 1)/2) + 1)/7 is prime.

Original entry on oeis.org

5, 31, 53, 163, 509, 1061, 13627, 20047, 28411, 50993, 71453, 272141, 1353449
Offset: 1

Views

Author

Alexander Adamchuk, Dec 04 2006

Keywords

Comments

PrimePi[ a(n) ] = {3, 11, 16, 38, 97, 178,...}.

Links

Crossrefs

Cf. A125744 = Primes p such that (3^p + 3^((p + 1)/2) + 1)/7 is prime. Cf. A125738 = Primes p such that 3^p - 3^((p + 1)/2) + 1 is prime. Cf. A125739 = Primes p such that 3^p + 3^((p + 1)/2) + 1 is prime. Cf. A007670, A007671, A124165, A125742.

Programs

  • Mathematica
    Do[p=Prime[n];f=(3^p-3^((p+1)/2)+1)/7;If[PrimeQ[f],Print[{n,p}]],{n,1,178}]

Extensions

a(6)-a(11) from Lelio R Paula (lelio(AT)sknet.com.br), May 08 2008
a(12) from Serge Batalov, Mar 07 2014
a(13) from Serge Batalov, Mar 25 2014

A125744 Primes p such that (3^p + 3^((p + 1)/2) + 1)/7 is prime.

Original entry on oeis.org

11, 37, 47, 97, 167, 877, 2027, 2293, 3011, 6803, 8423, 50221, 152809, 505823
Offset: 1

Views

Author

Alexander Adamchuk, Dec 04 2006

Keywords

Comments

PrimePi[ a(n) ] = {5, 12, 15, 25, 39, 151, 307, 341, ...}.

Links

Crossrefs

Cf. A125743 = Primes p such that (3^p - 3^((p + 1)/2) + 1)/7 is prime. Cf. A125738 = Primes p such that 3^p - 3^((p + 1)/2) + 1 is prime. Cf. A125739 = Primes p such that 3^p + 3^((p + 1)/2) + 1 is prime. Cf. A007670, A007671, A124165, A125742.

Programs

  • Mathematica
    Do[p=Prime[n];f=(3^p+3^((p+1)/2)+1)/7;If[PrimeQ[f],Print[{n,p}]],{n,1,341}]

Extensions

a(9)-a(12) from Lelio R Paula (lelio(AT)sknet.com.br), May 09 2008
a(13)-a(14) from Serge Batalov, Mar 07 2014
Showing 1-6 of 6 results.

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