A028491 -id:A028491 - cp's OEIS frontend

A076481 Primes of the form (3^n-1)/2.

13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013

Offset: 1

Dean Hickerson, Oct 14 2002

nonn

All primes p whose reciprocals belong to the middle-third Cantor set satisfy an equation of the form 2pK + 1 = 3^n. This sequence is the special case K = 1. See reference. [Christian Salas, Jul 04 2011]

Conjecture: primes p such that sigma(2p+1) = 3*p+1. Sigma(2*a(n)+1) = 3*a(n) +1 holds for all first 9 terms. - Jaroslav Krizek, Sep 28 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..9
Christian Salas, On prime reciprocals in the Cantor set, arXiv:0906.0465v5 [math.NT], 2009-2011.
Christian Salas, Cantor primes as prime-valued cyclotomic polynomials, arXiv preprint arXiv:1203.3969 [math.NT], 2012.

The exponents n are in A028491. Cf. A075081.

[a: n in [1..200] | IsPrime(a) where a is (3^n-1) div 2 ]; // Vincenzo Librandi, Dec 09 2011

A076481:=n->`if`(isprime((3^n-1)/2), (3^n-1)/2, NULL): seq(A076481(n), n=1..100); # Wesley Ivan Hurt, Sep 30 2014

Select[Table[(3^n-1)/2, {n,0,500}], PrimeQ] (* Vincenzo Librandi, Dec 09 2011 *)

for(n=3,99,if(ispseudoprime(t=3^n\2),print1(t", "))) \\ Charles R Greathouse IV, Jul 02 2013

A004063 Numbers k such that (7^k - 1)/6 is prime.

Original entry on oeis.org

5, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699

Offset: 1

N. J. A. Sloane

Base-7 repunit primes. - Paul Bourdelais, Aug 31 2007

Among repunits with bases from -11 to 11, base-7 repunits have the lowest relative rate of occurrence of primes so far. - Paul Bourdelais, Feb 23 2010

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 236.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Paul Bourdelais, A Generalized Repunit Conjecture
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
H. Lifchitz, Mersenne and Fermat primes field
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Repunit

For[n = 1, n <= 20000, n++, If[PrimeQ[(7^n - 1)/6 ], Print[n]]] (* Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 09 2006 *)

is(n)=isprime((7^n - 1)/6) \\ Charles R Greathouse IV, Apr 28 2015

Prime95
```
PRP=1,7,1264699,-1,0,0,"6"
```

a(6) from Robert G. Wilson v, Apr 09 2005
a(7) is a probable prime from Paul Bourdelais, Aug 31 2007
a(8) discovered Sep 17 2008 by Paul Bourdelais & Eric Purohit - it is a probable prime based on trial factoring to 2.5*10^13 and Fermat base 2 primality test. - Paul Bourdelais, Sep 18 2008
a(9) is a probable prime discovered by Paul Bourdelais, Feb 23 2010
a(10) is a probable prime discovered by Paul Bourdelais, Jan 06 2014

A005808 Numbers k such that (11^k - 1)/10 is prime.

Original entry on oeis.org

17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831, 1868983

Offset: 1

N. J. A. Sloane

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 236.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. Bourdelais, A Generalized Repunit Conjecture
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
H. Lifchitz, Mersenne and Fermat primes field
Henri & Renaud Lifchitz, PRP Records.
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Repunit.
Index to primes in various ranges, form ((k+1)^n-1)/k

lst={};Do[If[PrimeQ[(11^n-1)/10], Print[n];AppendTo[lst, n]], {n, 10^5}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)

is(n)=ispseudoprime((11^n-1)/10) \\ Charles R Greathouse IV, Apr 29 2015

a(11) = 20161 was found by Kamil Duszenko on Aug 15 2003. - Alexander Adamchuk, Feb 11 2007
a(12) = 293831 corresponds to a probable prime discovered by Paul Bourdelais with PFGW v3.3.1, Mar 08 2010
a(13) by Paul Bourdelais, Jun 01 2021

A004062 Numbers k such that (6^k - 1)/5 is prime.

Original entry on oeis.org

2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099, 1365019, 3360347

Offset: 1

N. J. A. Sloane

Prime repunits in base 6.
With this 16th prime, the base 6 repunits have an average (best linear fit) occurrence rate of G = 0.4948 which seems to be converging to the conjectured rate of 0.56146 (see ref).
Also, numbers k such that 6^k-1 is semiprime. - Sean A. Irvine, Oct 16 2023

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 236.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Brillhart et al., Cunningham Project [Factorizations of b^n +- 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers]
Paul Bourdelais, A Generalized Repunit Conjecture. - _Paul Bourdelais_, May 24 2010
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
H. Lifchitz, Mersenne and Fermat primes field
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Repunit
Index to primes in various ranges, form ((k+1)^n-1)/k

Select[Range[1000], PrimeQ[(6^# - 1)/5] &] (* Alonso del Arte, Dec 31 2019 *)

is(n)=isprime((6^n - 1)/5) \\ Charles R Greathouse IV, Apr 28 2015

More terms from Kamil Duszenko (kdusz(AT)wp.pl), Jun 22 2003
a(14) discovered Nov 05 2007, corresponds to a probable prime based on trial factoring to 10^11 and Fermat primality test base 2. - Paul Bourdelais
a(15) corresponds to a probable prime discovered by Paul Bourdelais, May 24 2010
a(16) corresponds to a probable prime discovered by Paul Bourdelais, Dec 31 2019
a(17) corresponds to a probable prime discovered by Ryan Propper, Oct 30 2023

A004064 Numbers k such that (12^k - 1)/11 is prime.

Original entry on oeis.org

2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889, 769543

Offset: 1

N. J. A. Sloane

Also, numbers k such that 12^k-1 is a semiprime. - Sean A. Irvine, Oct 16 2023

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 109, p. 38, Ellipses, Paris 2008.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 236.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. Bourdelais, A Generalized Repunit Conjecture
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
H. Lifchitz, Mersenne and Fermat primes field
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Repunit

lst={}; Do[If[PrimeQ[(12^n-1)/11], Print[n]; AppendTo[lst, n]], {n, 10^5}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)

is(n)=ispseudoprime((12^n-1)/11) \\ Charles R Greathouse IV, Apr 29 2015

a(11) from Paul Bourdelais, Aug 03 2007
One more term from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 05 2008
a(13)=46889, discovered Sep 10 2008 by Paul Bourdelais, corresponds to a probable prime based on trial factoring to 10^13 and Fermat base 2 primality test. - Paul Bourdelais, Sep 11 2008
a(14)=769543 corresponds to a probable prime discovered by Paul Bourdelais, Dec 05 2014

A016054 Numbers n such that (13^n - 1)/12 is prime.

Original entry on oeis.org

5, 7, 137, 283, 883, 991, 1021, 1193, 3671, 18743, 31751, 101089, 1503503

Offset: 1

N. J. A. Sloane

nonn

For Repunits in bases from -14 to 14, base 13 is a lucky number with the highest relative rate of primes being discovered. Base 7 is the most unlucky base with the lowest rate of primes being discovered. There is a Generalized Repunit Conjecture implying that all bases will eventually converge to the same relative rate of occurrence (ref 1). - Paul Bourdelais, Mar 01 2010

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. Bourdelais, A Generalized Repunit Conjecture
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
H. Lifchitz, Mersenne and Fermat primes field

lst={};Do[If[PrimeQ[(13^n-1)/12], Print[n];AppendTo[lst, n]], {n, 10^5}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)

is(n)=isprime((13^n-1)/12) \\ Charles R Greathouse IV, Feb 17 2017

Error in first term corrected by Robert G. Wilson v, Aug 15 1997
a(10) (corresponds to a probable prime) from David Radcliffe, Jul 04 2004
a(11) from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 05 2008
a(12) corresponds to a probable prime discovered by Paul Bourdelais, Mar 01 2010
a(13) corresponds to a probable prime discovered by Paul Bourdelais, Apr 09 2020

A006033 Numbers n such that (15^n - 1)/14 is prime.

Original entry on oeis.org

3, 43, 73, 487, 2579, 8741, 37441, 89009, 505117, 639833

Offset: 1

N. J. A. Sloane

8741 and 37441 are only probable primes. - Julien Peter Benney (jpbenney(AT)ftml.net), Apr 27 2007

			(15^3 - 1)/14 = 241, which is prime.

Paulo Ribenboim, "The Book Of Prime Number Records"; published 1989 by Springer-Verlag; pages 350-354.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. Bourdelais, A Generalized Repunit Conjecture
Harvey Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
Henri Lifchitz, Mersenne and Fermat primes field
Index to primes in various ranges, form ((k+1)^n-1)/k

Cf. A059802, A062647, A003525.

lst={};Do[If[PrimeQ[(15^n-1)/14], Print[n];AppendTo[lst, n]], {n, 10^5}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)

is(n)=ispseudoprime((15^n-1)/14) \\ Charles R Greathouse IV, Apr 29 2015

a(7) from Julien Peter Benney (jpbenney(AT)ftml.net), Apr 27 2007
a(8) corresponds to a probable prime discovered by Paul Bourdelais, Mar 15 2010
a(9) corresponds to a probable prime discovered by Paul Bourdelais, Jan 14 2015
a(10) corresponds to a probable prime discovered by Paul Bourdelais, Apr 22 2019

A006032 Numbers k such that (14^k - 1)/13 is prime.

Original entry on oeis.org

3, 7, 19, 31, 41, 2687, 19697, 59693, 67421, 441697

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Paul Bourdelais, A Generalized Repunit Conjecture
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
H. Lifchitz, Mersenne and Fermat primes field

lst={};Do[If[PrimeQ[(14^n-1)/13], Print[n];AppendTo[lst, n]], {n, 10^5}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)

is(n)=isprime((14^n - 1)/13) \\ Charles R Greathouse IV, Apr 29 2015

One more term from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 05 2008
a(8) and a(9) correspond to probable primes discovered by Paul Bourdelais, Mar 01 2010
a(10) corresponds to a probable prime discovered by Paul Bourdelais, Dec 08 2014

A006034 Numbers n such that (17^n-1)/16 is prime.

Original entry on oeis.org

3, 5, 7, 11, 47, 71, 419, 4799, 35149, 54919, 74509, 1990523

Offset: 1

N. J. A. Sloane

No others for any n less than 8447. - Julien Peter Benney (jpbenney(AT)ftml.net), Aug 15 2004

Ribenboim, Paulo; "The Book Of Prime Number Records"; published 1989 by Springer-Verlag; pages 350-354.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
H. Lifchitz, Mersenne and Fermat primes field

lst={};Do[If[PrimeQ[(17^n-1)/16], Print[n];AppendTo[lst, n]], {n, 10^5}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)

is(n)=isprime((17^n-1)/16) \\ Charles R Greathouse IV, Apr 28 2015

a(9)=35149 & a(10)=54919 are probable primes discovered by Paul Bourdelais, Mar 08 2010
a(11)=74509 is a probable prime discovered by Paul Bourdelais, Mar 10 2010
a(12)=1990523 corresponds to a probable prime discovered by Paul Bourdelais, Aug 03 2020

A006035 Numbers n such that (19^n-1)/18 is prime.

Original entry on oeis.org

19, 31, 47, 59, 61, 107, 337, 1061, 9511, 22051, 209359

Offset: 1

N. J. A. Sloane

No others less than 8011. - Julien Peter Benney (jpbenney(AT)ftml.net), Aug 15 2004

a(9) = 9511 was found by Richard Fischer on Dec 15 2004. - Alexander Adamchuk, Feb 11 2007

Ribenboim, Paulo; "The Book Of Prime Number Records"; published 1989 by Springer-Verlag; pages 350-354.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Paul Bourdelais, A Generalized Repunit Conjecture. - _Paul Bourdelais_, Aug 27 2010
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
H. Lifchitz, Mersenne and Fermat primes field
Index to primes in various ranges, form ((k+1)^n-1)/k

lst={};Do[If[PrimeQ[(19^n-1)/18], Print[n];AppendTo[lst, n]], {n, 10^5}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)

is(n)=isprime((19^n-1)/18) \\ Charles R Greathouse IV, Apr 28 2015

One more term from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 05 2008

a(11)=209359 corresponds to a probable prime discovered by Paul Bourdelais, Aug 27 2010

cp's OEIS Frontend

A076481 Primes of the form (3^n-1)/2.

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A004063 Numbers k such that (7^k - 1)/6 is prime.

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A005808 Numbers k such that (11^k - 1)/10 is prime.

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A004062 Numbers k such that (6^k - 1)/5 is prime.

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A004064 Numbers k such that (12^k - 1)/11 is prime.

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A016054 Numbers n such that (13^n - 1)/12 is prime.

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A006033 Numbers n such that (15^n - 1)/14 is prime.

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