A057429 -id:A057429 - cp's OEIS frontend

A172512 Partial sums of A057429.

2, 5, 10, 17, 28, 47, 76, 123, 196, 275, 388, 539, 696, 859, 1026, 1265, 1506, 1789, 2142, 2509, 2888, 3345, 4342, 5709, 8750, 18891, 33590, 61119, 110326, 187617, 272854, 379547, 539970, 743759, 1108048, 2100009, 3303802, 4971123, 8675176, 13467233, 28784460

Offset: 1

Jonathan Vos Post, Feb 05 2010

nonn

Cf. A057429.

More terms from the data at A057429 added by Amiram Eldar, Oct 19 2024

A000043 Mersenne exponents: primes p such that 2^p - 1 is prime. Then 2^p - 1 is called a Mersenne prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281

Offset: 1

N. J. A. Sloane

Equivalently, integers k such that 2^k - 1 is prime.
It is believed (but unproved) that this sequence is infinite. The data suggest that the number of terms up to exponent N is roughly K log N for some constant K.
Length of prime repunits in base 2.
The associated perfect number N=2^(p-1)*M(p) (=A019279*A000668=A000396), has 2p (=A061645) divisors with harmonic mean p (and geometric mean sqrt(N)). - Lekraj Beedassy, Aug 21 2004
In one of his first publications Euler found the numbers up to 31 but erroneously included 41 and 47.
Equals number of bits in binary expansion of n-th Mersenne prime (A117293). - Artur Jasinski, Feb 09 2007
Number of divisors of n-th even perfect number, divided by 2. Number of divisors of n-th even perfect number that are powers of 2. Number of divisors of n-th even perfect number that are multiples of n-th Mersenne prime A000668(n). - Omar E. Pol, Feb 24 2008
Number of divisors of n-th even superperfect number A061652(n). Numbers of divisors of n-th superperfect number A019279(n), assuming there are no odd superperfect numbers. - Omar E. Pol, Mar 01 2008
Differences between exponents when the even perfect numbers are represented as differences of powers of 2, for example: The 5th even perfect number is 33550336 = 2^25 - 2^12 then a(5)=25-12=13 (see A135655, A133033, A090748). - Omar E. Pol, Mar 01 2008
Number of 1's in binary expansion of n-th even perfect number (see A135650). Number of 1's in binary expansion of divisors of n-th even perfect number that are multiples of n-th Mersenne prime A000668(n) (see A135652, A135653, A135654, A135655). - Omar E. Pol, May 04 2008
Indices of the numbers A006516 that are also even perfect numbers. - Omar E. Pol, Aug 30 2008
Indices of Mersenne numbers A000225 that are also Mersenne primes A000668. - Omar E. Pol, Aug 31 2008
The (prime) number p appears in this sequence if and only if there is no prime q<2^p-1 such that the order of 2 modulo q equals p; a special case is that if p=4k+3 is prime and also q=2p+1 is prime then the order of 2 modulo q is p so p is not a term of this sequence. - Joerg Arndt, Jan 16 2011
Primes p such that sigma(2^p) - sigma(2^p-1) = 2^p-1. - Jaroslav Krizek, Aug 02 2013
Integers k such that every degree k irreducible polynomial over GF(2) is also primitive, i.e., has order 2^k-1.  Equivalently, the integers k such that A001037(k) = A011260(k). - Geoffrey Critzer, Dec 08 2019
Conjecture: for k > 1, 2^k-1 is (a Mersenne) prime or k = 2^(2^m)+1 (is a Fermat number) if and only if (k-1)^(2^k-2) == 1 (mod (2^k-1)k^2). - Thomas Ordowski, Oct 05 2023
Conjecture: for p prime, 2^p-1 is (a Mersenne) prime or p = 2^(2^m)+1 (is a Fermat number) if and only if (p-1)^(2^p-2) == 1 (mod 2^p-1). - David Barina, Nov 25 2024
Already as of Dec. 2020, all exponents up to 10^8 had been verified, implying that 74207281, 77232917 and 82589933 are indeed the next three terms. As of today, all exponents up to 130439863 have been tested at least once, see the GIMPS Milestones Report. - M. F. Hasler, Apr 11 2025
On June 23. 2025 all exponents up to 74340751 have been verified, confirming that 74207281 is the exponent of the 49th Mersenne Prime. - Rodolfo Ruiz-Huidobro, Jun 23 2025

			Corresponding to the initial terms 2, 3, 5, 7, 13, 17, 19, 31 ... we get the Mersenne primes 2^2 - 1 = 3, 2^3 - 1 = 7, 2^5 - 1 = 31, 127, 8191, 131071, 524287, 2147483647, ... (see A000668).

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 4.
J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, p. 79.
R. K. Guy, Unsolved Problems in Number Theory, Section A3.
F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 2000, p. 57.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 19.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 47.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 132-134.
B. Tuckerman, The 24th Mersenne prime, Notices Amer. Math. Soc., 18 (Jun, 1971), Abstract 684-A15, p. 608.

P. T. Bateman, J. L. Selfridge, and S. S. Wagstaff, Jr., The new Mersenne conjecture, Amer. Math. Monthly 96 (1989), no. 2, 125--128. MR0992073 (90c:11009).
J. Bernheiden, Mersenne Numbers (Text in German)
Andrew R. Booker, The Nth Prime Page
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
P. G. Brown, A Note on Ramanujan's (FALSE) Conjectures Regarding 'Mersenne Primes'
C. K. Caldwell, Mersenne Primes
C. K. Caldwell, Recent Mersenne primes
Zuling Chang, Martianus Frederic Ezerman, Adamas Aqsa, Fahreza, San Ling, Janusz Szmidt, and Huaxiong Wang, Binary de Bruijn Sequences via Zech's Logarithms, 2018.
Keith Conrad, Square patterns and infinitude of primes, University of Connecticut, 2019.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
Leonhard Euler, Observations on a theorem of Fermat and others on looking at prime numbers, arXiv:math/0501118 [math.HO], 2005-2008.
Leonhard Euler, Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus
G. Everest et al., Primes generated by recurrence sequences, arXiv:math/0412079 [math.NT], 2006.
G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
F. Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Luis H. Gallardo and Olivier Rahavandrainy, On (unitary) perfect polynomials over F_2 with only Mersenne primes as odd divisors, arXiv:1908.00106 [math.NT], 2019.
Donald B. Gillies, Three new Mersenne primes and a statistical theory Mathematics of Computation 18.85 (1964): 93-97.
GIMPS (Great Internet Mersenne Prime Search), Distributed Computing Projects
GIMPS, Milestones Report
GIMPS, GIMPS Project discovers largest known prime number 2^77232917-1
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
H. Lifchitz, Mersenne and Fermat primes field
A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, 1996; see p. 143.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012.
Romeo Meštrović, Goldbach-type conjectures arising from some arithmetic progressions, University of Montenegro, 2018.
Romeo Meštrović, Goldbach's like conjectures arising from arithmetic progressions whose first two terms are primes, arXiv:1901.07882 [math.NT], 2019.
G. P. Michon, Perfect Numbers, Mersenne Primes
Albert A. Mullin, Letter to the editor, about "The new Mersenne conjecture" [Amer. Math. Monthly 96 (1989), no. 2, 125-128; MR0992073 (90c:11009)] by P. T. Bateman, J. L. Selfridge and S. S. Wagstaff, Jr., Amer. Math. Monthly 96 (1989), no. 6, 511. MR0999415 (90f:11008).
Curt Noll and Laura Nickel, The 25th and 26th Mersenne primes, Math. Comp. 35 (1980), 1387-1390.
M. Oakes, A new series of Mersenne-like Gaussian primes
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.
H. J. Smith, Mersenne Primes
B. Tuckerman, The 24th Mersenne prime, Proc. Nat. Acad. Sci. USA, 68 (1971), 2319-2320.
H. S. Uhler, On All Of Mersenne's Numbers Particularly M_193, PNAS 1948 34 (3) 102-103.
H. S. Uhler, First Proof That The Mersenne Number M_157 Is Composite, PNAS 1944 30(10) 314-316.
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Cunningham Number
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Mersenne Prime
Eric Weisstein's World of Mathematics, Repunit
Eric Weisstein's World of Mathematics, Wagstaff's Conjecture
David Whitehouse, Number takes prime position (2^13466917 - 1 found after 13000 years of computer time)
K. Zsigmondy, Zur Theorie der Potenzreste, Monatshefte für Mathematik und Physik, Vol. 3, No. 1 (1892), 265-284.
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime
Index entries for "core" sequences

Cf. A000668 (Mersenne primes).
Cf. A028335 (integer lengths of Mersenne primes).
Cf. A000225 (Mersenne numbers).
Cf. A001348 (Mersenne numbers with n prime).
Cf. A016027, A046051, A057429, A057951-A057958, A066408, A117293, A127962, A127963, A127964, A127965, A127961, A000979, A000978, A124400, A124401, A127955, A127956, A127957, A127958, A127936, A134458, A000225, A000396, A090748, A133033, A135655, A006516, A019279, A061652, A133033, A135650, A135652, A135653, A135654, A260073, A050475, A379590.

MersennePrimeExponent[Range[48]] (* Eric W. Weisstein, Jul 17 2017; updated Oct 21 2024 *)

isA000043(n) = isprime(2^n-1) \\ Michael B. Porter, Oct 28 2009

is(n)=my(h=Mod(2,2^n-1)); for(i=1, n-2, h=2*h^2-1); h==0||n==2 \\ Lucas-Lehmer test for exponent e. - Joerg Arndt, Jan 16 2011, and Charles R Greathouse IV, Jun 05 2013
forprime(e=2,5000,if(is(e),print1(e,", "))); /* terms < 5000 */

from sympy import isprime, prime
for n in range(1,100):
    if isprime(2**prime(n)-1):
        print(prime(n), end=', ') # Stefano Spezia, Dec 06 2018

a(n) = log((1/2)*(1+sqrt(1+8*A000396(n))))/log(2). - Artur Jasinski, Sep 23 2008 (under the assumption there are no odd perfect numbers, Joerg Arndt, Feb 23 2014)
a(n) = A000005(A061652(n)). - Omar E. Pol, Aug 26 2009
a(n) = A000120(A000396(n)), assuming there are no odd perfect numbers. - Omar E. Pol, Oct 30 2013

Also in the sequence: p = 74207281. - Charles R Greathouse IV, Jan 19 2016
Also in the sequence: p = 77232917. - Eric W. Weisstein, Jan 03 2018
Also in the sequence: p = 82589933. - Gord Palameta, Dec 21 2018
a(46) = 42643801 and a(47) = 43112609, whose ordinal positions in the sequence are now confirmed, communicated by Eric W. Weisstein, Apr 12 2018
a(48) = 57885161, whose ordinal position in the sequence is now confirmed, communicated by Benjamin Przybocki, Jan 05 2022
Also in the sequence: p = 136279841. - Eric W. Weisstein, Oct 21 2024
As of Jan 31 2025, 48 terms are known, and are shown in the DATA section. Four additional numbers are known to be in the sequence, namely 74207281, 77232917, 82589933, and 136279841, but they may not be the next terms. See the GIMP website for the latest information. - N. J. A. Sloane, Jan 31 2025

A066408 Numbers n such that the Eisenstein integer (1 - ω)^n - 1 has prime norm, where ω = -1/2 + sqrt(-3)/2.

Original entry on oeis.org

2, 5, 7, 11, 17, 19, 79, 163, 193, 239, 317, 353, 659, 709, 1049, 1103, 1759, 2029, 5153, 7541, 9049, 10453, 23743, 255361, 534827, 2237561, 2888387, 4043119

Offset: 1

Mike Oakes, Dec 24 2001

Analog of Mersenne primes in Eisenstein integers.
The norm of a + b * ω is (a + b * ω) * (a + b * ω^2) = a^2 + a*b + b^2.
Indices for which the Eisenstein-Mersenne numbers are primes. The p-th Eisenstein-Mersenne number can be written as 3^p - Legendre(3, p) * 3^((p + 1)/2) + 1. Note the enormous gap between 23743 and 255361. A modified version of Chris Nash's PFGW program was used to find the last term. - Jeroen Doumen (doumen(AT)win.tue.nl), Oct 31 2002
Let q be the integer quaternion (3 + i + j + k)/2. Then q^n - 1 is a quaternion prime for these n; that is, the norm of q^n - 1 is a rational prime. - T. D. Noe, Feb 02 2005
The actual norms also belong to the class of Generalized Unique primes (see Links section), that is primes which have a period of expansion of 1/p (in some general, non-decimal system) that it shares with no other prime. - Serge Batalov, Mar 29 2014
Next term > 4400000. - Serge Batalov, Jun 20 2023

			For n = 7, (1 - ω)^7 - 1 has norm 2269, a prime.
Or, for p = 7, 3^7 + 3^4 + 1 = 2269, which is prime.

P. H. T. Beelen, Algebraic geometry and coding theory, Ph.D. Thesis, Eindhoven, The Netherlands, September 2001.
J. M. Doumen, Ph.D. Thesis, Eindhoven, The Netherlands, to appear.
Mike Oakes, posting to primenumbers(AT)yahoogroups.com, Dec 24 2001.

Pedro Berrizbeitia and Boris Iskra, Gaussian Mersenne and Eisenstein Mersenne primes, Mathematics of Computation 79 (2010), pp. 1779-1791.
Chris Caldwell, The largest known primes
Chris Caldwell, Generalized Unique primes
Mike Oakes, Eisenstein Mersenne and Fermat primes
Mike Oakes, Eisenstein Mersenne and Fermat primes, message 4607 in primenumbers Yahoo group, Dec 24, 2001.
Mike Oakes, A new series of Mersenne-like Gaussian primes
Mike Oakes, Posting to the Number Theory list, Dec 27 2005.
K. Pershell and L. Huff, Mersenne Primes in Imaginary Quadratic Number Fields, (2002).
Eric Weissteins's World of Mathematics, Eisenstein Integer

The actual norms are in A066413.

Cf. A000043, A010527, A057429, A125738, A125739.

maxPi = 3000; primeNormQ[p_] := PrimeQ[1 + 3^p - 2*3^(p/2)*Cos[(p*Pi)/6]]; A066408 = {}; Do[ If[primeNormQ[p = Prime[k]], Print[p]; AppendTo[A066408, p]], {k, 1, maxPi}]; A066408 (* Jean-François Alcover, Oct 21 2011 *)

print1("2, "); /*the only even member; it is special*/ forprime(n=3,2029,if(ispseudoprime(3^n-kronecker(3,n)*3^((n+1)/2)+1),print1(n, ", "))) \\ Serge Batalov, Mar 29 2014

a(26) from Serge Batalov, Mar 29 2014
a(27) from Ryan Propper and Serge Batalov, Jun 18 2023
a(28) from Ryan Propper and Serge Batalov, Jun 20 2023
Corrected link to NMBRTHRY posting. - Serge Batalov, Apr 01 2014

A007670 Numbers n such that 2^n - 2^((n + 1)/2) + 1 is prime.

Original entry on oeis.org

3, 7, 47, 73, 79, 113, 151, 167, 239, 241, 353, 367, 457, 1367, 3041, 27529, 49207, 160423, 364289, 991961, 1203793, 1667321, 4792057

Offset: 1

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

If A007670 is a proper subset of A057429, then 364289 & 991961 are the next two terms.

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).

J. Brillhart, Concerning the number 2^(2p)+1, p prime, Math. Comp. 16 (80) (1962) 424-430.
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A006598.

is(n)=isprime(2^n-2^((n+1)/2)+1) \\ Charles R Greathouse IV, Feb 17 2017

a(16)-a(18) from Robert G. Wilson v, Sep 07 2000

a(19)-a(23) from Serge Batalov, Jun 16 2020

A007671 Numbers n such that 2^n + 2^((n + 1)/2) + 1 is prime.

Original entry on oeis.org

1, 3, 5, 11, 19, 29, 157, 163, 283, 379, 997, 10141, 14699, 77291, 85237, 106693, 203789, 3704053

Offset: 1

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

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).

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A057429.

for(n=1, 2000, if(n%2==1&&ispseudoprime(2^n+2^((n+1)/2)+1), print1(n,","))) \\ Charles R Greathouse IV, Feb 17 2017 edited by Aurelien Gibier Jun 29 2024.

More terms from Robert G. Wilson v, Sep 07 2000
203789 found and proved prime by Mike Oakes, on Sep 28 2000.
a(18) from Serge Batalov, Jun 16 2020

A182300 Gaussian-Mersenne primes: primes of the form ((1+i)^p - 1)((1-i)^p - 1).

Original entry on oeis.org

5, 13, 41, 113, 2113, 525313, 536903681, 140737471578113, 9444732965601851473921, 604462909806215075725313, 10384593717069655112945804582584321, 2854495385411919762116496381035264358442074113

Offset: 1

Arkadiusz Wesolowski, Apr 23 2012

See A057429 for the values of p.

Primes of the form q = 2^p +- 2^((p+1)/2) + 1. Note that q == 1 (mod p). - Thomas Ordowski, Apr 18 2019

John Brillhart et al., Factorizations of b^n +/- 1, b=2,3,5,6,7,10,12 up to high powers, Amer. Math. Soc., Providence RI, 1988, pp. xcvi+236.
R. K. Guy, Unsolved Problems in Number Theory, New York: Springer-Verlag, 1994, pp. 33-36.
Miriam Hausmann and Harold N. Shapiro, Perfect Ideals over the Gaussian Integers, Comm. Pure Appl. Math. 29 (1976), pp. 323-341.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..25
Bogley, William A.; Williams, Gerald Efficient finite groups arising in the study of relative asphericity. Math. Z. 284, No. 1-2, 507-535 (2016).
Chris Caldwell, The Prime Glossary, Gaussian Mersenne
C. K. Caldwell, "Top Twenty" page, Gaussian Mersenne norm
Ellen Gethner, Stan Wagon, and Brian Wick, A Stroll Through the Gaussian Primes, Amer. Math. Monthly 105 (1998), pp. 327-337.
W. L. McDaniel, Perfect Gaussian integers, Acta Arithmetica 25 (1974), pp. 137-144.
Index entries for Gaussian integers and primes

Cf. A088962, A057429.

lst = {}; Do[a = (1 + I)^n - 1; b = a*Conjugate[a]; If[PrimeQ[b], AppendTo[lst, b]], {n, 151}]; lst
gmp[n_]:=Module[{x=(1+I)^n-1},x*Conjugate[x]]; Select[Table[gmp[n],{n,200}],PrimeQ] (* Harvey P. Dale, Apr 27 2016 *)

A027206 Numbers m such that (1+i)^m + i is a Gaussian prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 11, 14, 16, 19, 38, 47, 62, 79, 151, 163, 167, 214, 239, 254, 283, 367, 379, 1214, 1367, 2558, 4406, 8846, 14699, 49207, 77291, 160423, 172486, 221006, 432182, 1513678, 2515574

Offset: 1

Ed Pegg Jr, Aug 07 2002

nonn

Equivalently, either (1+i)^m + i times its conjugate is an ordinary prime, or m == 2 (mod 4) and 2^(m/2) + (-1)^((m-2)/4) is an ordinary prime.

Let z = (1+i)^m + i. If z is not pure real or pure imaginary, then z is a Gaussian prime if the product of z and its conjugate is a rational prime. That product is 1 + 2^m + sin(m*Pi/4)*2^(1+m/2). z is imaginary when m=4k+2, in which case z has magnitude 2^(2k+1) + (-1)^k. These pure imaginary numbers are Gaussian primes when 2^(2k+1)-1 is a Mersenne prime and 2k+1 == 1 (mod 4); that is, when m is twice an odd number in A112633. - T. D. Noe, Mar 07 2011

Index entries for Gaussian integers and primes

Cf. A057429, A103329.

Select[Range[0,30000], PrimeQ[(1+I)^#+I, GaussianIntegers->True]&]

More terms from Mike Oakes, Aug 07 2002
Edited by Dean Hickerson, Aug 14 2002
0 prepended by T. D. Noe, Mar 07 2011

A088962 Values of n that generate Generalized Gaussian-Mersenne primes (see below).

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 10, 11, 12, 14, 15, 18, 19, 21, 22, 26, 27, 29, 30, 33, 34, 35, 42, 45, 47, 49, 51, 54, 55, 58, 63, 65, 66, 69, 70, 73, 79, 85, 86, 87, 105, 106, 110, 111, 113, 114, 126, 129, 138, 147, 151, 157, 163, 167, 178, 186, 189, 217, 231, 239, 241, 242, 283

Offset: 1

Marc Chamberland, Oct 28 2003

nonn

M. Chamberland, Binary BBP-Formulae for Logarithms and Generalized Gaussian-Mersenne Primes, Journal of Integer Sequences, v.6 (2003), article 03.3.7, 1-10.

Cf. A057429, A182300 (Gaussian-Mersenne primes).

t = {}; Do[s = FullSimplify[Exp[2 Re[Log [Cyclotomic[n, (1 + I)/2]]]]]; If[PrimeQ[Numerator[s]], AppendTo[t, n]], {n, 100}]; t (* T. D. Noe, May 02 2012 *)

The numerator of the rational expression exp(2*Re(log(Phi_n((1+i)/2)))) is prime, where Phi_n is the n-th cyclotomic polynomial.

A103329 Numbers n such that (1+i)^n - i is a Gaussian prime.

Original entry on oeis.org

0, 3, 4, 5, 8, 10, 16, 26, 29, 34, 73, 113, 122, 157, 178, 241, 353, 457, 997, 1042, 3041, 4562, 6434, 8506, 10141, 19378, 19882, 22426, 27529

Offset: 1

T. D. Noe, Jan 31 2005

nonn

Note that A027206 and A057429 treat Gaussian primes of a similar form. The remaining case, (1+i)^n + 1, is a Gaussian prime for n=1,2,3,4 only.

Let z = (1+i)^n - i. If z is not pure real or pure imaginary, then z is a Gaussian prime if the product of z and its conjugate is a rational prime. That product is 1 + 2^n - sin(n*Pi/4)*2^(1+n/2). z is real when n=1. z is imaginary when n=4k+2, in which case, z has magnitude 2^(2k+1) - (-1)^k. These pure imaginary numbers are Gaussian primes when 2^(2k+1)-1 is a Mersenne prime and 2k+1 = 3 (mod 4); that is, when n is twice an odd number in A112634. - T. D. Noe, Mar 07 2011

Cf. A027206 ((1+i)^n + i is a Gaussian prime), A057429 ((1+i)^n - 1 is a Gaussian prime).

fQ[n_] := PrimeQ[(1 + I)^n - I, GaussianIntegers -> True]; Select[ Range[0, 30000], fQ]

a(25)-a(29) from Robert G. Wilson v, Mar 02 2011.

0 prepended by T. D. Noe, Mar 07 2011

A207040 Generalized Gaussian-Mersenne primes (see below).

Original entry on oeis.org

5, 13, 29, 37, 41, 61, 109, 113, 397, 1321, 1429, 1613, 2113, 14449, 26317, 246241, 279073, 312709, 525313, 4327489, 7416361, 29247661, 47392381, 107367629, 536903681, 1326700741, 40388473189, 118750098349, 275415303169, 415878438361, 1759217765581

Offset: 1

Arkadiusz Wesolowski, May 07 2012

nonn

Marc Chamberland, Binary BBP-Formulae for Logarithms and Generalized Gaussian-Mersenne Primes, Journal of Integer Sequences, Vol. 6 (2003), Article 03.3.7.
Index entries for Gaussian integers and primes

Supersequence of A182300 (Gaussian-Mersenne primes). Cf. A088962, A057429.

lst = {}; Do[s = Numerator@FullSimplify@Exp[2*Re@Log@Cyclotomic[n, (1 + I)/2]]; If[PrimeQ[s] && ! MemberQ[lst, s], AppendTo[lst, s]], {n, 2^7}]; Take[Sort[lst], 31]

The numerator of the rational expression exp(2*Re(log(Phi_n((1 + i)/2)))) is prime, where Phi_n is the n-th cyclotomic polynomial. See A088962 for the values of n that generate primes.

cp's OEIS Frontend

A172512 Partial sums of A057429.

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A000043 Mersenne exponents: primes p such that 2^p - 1 is prime. Then 2^p - 1 is called a Mersenne prime.

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A066408 Numbers n such that the Eisenstein integer (1 - ω)^n - 1 has prime norm, where ω = -1/2 + sqrt(-3)/2.

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A007670 Numbers n such that 2^n - 2^((n + 1)/2) + 1 is prime.

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A007671 Numbers n such that 2^n + 2^((n + 1)/2) + 1 is prime.

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A182300 Gaussian-Mersenne primes: primes of the form ((1+i)^p - 1)((1-i)^p - 1).

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A027206 Numbers m such that (1+i)^m + i is a Gaussian prime.

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A088962 Values of n that generate Generalized Gaussian-Mersenne primes (see below).

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