A056725 -id:A056725 - cp's OEIS frontend

A002235 Numbers m such that 3*2^m - 1 is prime.

0, 1, 2, 3, 4, 6, 7, 11, 18, 34, 38, 43, 55, 64, 76, 94, 103, 143, 206, 216, 306, 324, 391, 458, 470, 827, 1274, 3276, 4204, 5134, 7559, 12676, 14898, 18123, 18819, 25690, 26459, 41628, 51387, 71783, 80330, 85687, 88171, 97063, 123630, 155930, 164987, 234760

Offset: 1

N. J. A. Sloane

H. Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see pp. 381-384.
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).

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..68
C. K. Caldwell, The Prime Pages: Database Search Output
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
Mersenne Forum, 321 Search
H. Riesel, Lucasian criteria for the primality of N=h.2^n-1, Math. Comp., 23 (1969), 869-875.
H. Riesel, Lucasian criteria for the primality of N=h.2^n-1, Math. Comp., 23 (1969), 869-875. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Thabit ibn Kurrah Prime
Eric Weisstein's World of Mathematics, Thabit ibn Kurrah Rule
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A000043, A007505, A003307, A046865, A079906, A046866, A001771, A005541, A056725, A046867, A079907.

lst={};Do[If[PrimeQ[3*2^n-1], Print[n];AppendTo[lst, n]], {n, 10^5}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)

is(n)=ispseudoprime(3<Charles R Greathouse IV, Aug 27 2014

More terms from Eric W. Weisstein, Sep 29 2007
a(60) = 11484018 from The Prime Pages, from Pierre CAMI, Nov 25 2014
a(61)-a(62) from The Prime Pages, from Eric W. Weisstein, Nov 03 2015
Terms moved from Data to b-file, and more terms added to b-file, by Jeppe Stig Nielsen, Sep 07 2021

A003307 Numbers k such that 2*3^k - 1 is prime.

Original entry on oeis.org

1, 2, 3, 7, 8, 12, 20, 23, 27, 35, 56, 62, 68, 131, 222, 384, 387, 579, 644, 1772, 3751, 5270, 6335, 8544, 9204, 12312, 18806, 21114, 49340, 75551, 90012, 128295, 143552, 147488, 1010743, 1063844, 1360104

Offset: 1

N. J. A. Sloane

R. K. Guy, Unsolved Problems in Theory of Numbers, Section A3.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Kevin A. Broughan and Qizhi Zhou, On the Ratio of the Sum of Divisors and Euler's Totient Function II, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.2.
Steven Harvey, Williams Primes
W. Keller and J. Richstein, Solutions of the congruence a^(p-1) == 1 (mod p^r), Math. Comp. 74 (2005), 927-936.
H. C. Williams, The primality of certain integers of the form 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.
H. C. Williams and C. R. Zarnke, Some prime numbers of the forms 2*3^n+1 and 2*3^n-1, Math. Comp., 26 (1972), 995-998.

Cf. A002235, A046865, A079906, A046866, A001771, A005541, A056725, A046867, A079907.

Cf. A079363 (primes of the form 2*3^k - 1), A003306 (k such that 2*3^k + 1 is prime).

for(n=1,1e4,if(ispseudoprime(2*3^n-1),print1(n", "))) \\ Charles R Greathouse IV, Jul 16 2011

More terms from Douglas Burke (dburke(AT)nevada.edu)
More terms from T. D. Noe, Aug 24 2005
Corrected and extended by Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 05 2008
a(35) from Borys Jaworski, Sep 02 2011
a(36) from Borys Jaworski, Feb 13 2012
a(37) from Jeppe Stig Nielsen, Sep 28 2018

A001771 Numbers k such that 7*2^k - 1 is prime.

Original entry on oeis.org

1, 5, 9, 17, 21, 29, 45, 177, 18381, 22529, 24557, 26109, 34857, 41957, 67421, 70209, 169085, 173489, 177977, 363929, 372897

Offset: 1

N. J. A. Sloane

k is always of the form 4*j + 1.

If k is in the sequence and m=2^(k+2)*3*(7*2^k-1) then phi(m)+sigma(m)=3m (m is in the sequence A011251). The proof is easy. - Farideh Firoozbakht, Mar 04 2005

H. Riesel, "Prime numbers and computer methods for factorization", Progress in Mathematics, Vol. 57, Birkhäuser, Boston, 1985, Chap. 4, see pp. 381-384.
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).

Wilfrid Keller, List of primes k.2^n - 1 for k < 300
H. C. Williams and C. R. Zarnke, A report on prime numbers of the forms M = (6a+1)*2^(2m-1)-1 and (6a-1)*2^(2m)-1, Math. Comp., 22 (1968), 420-422.
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A050523, A003307, A002235, A046865, A079906, A046866, A005541, A056725, A046867, A079907.

Cf. A032353 (7*2^k+1 is prime).

Do[ If[ PrimeQ[7*2^n - 1], Print[n]], {n, 1, 2500}]

v=[ ]; for(n=0,2000, if(isprime(7*2^n-1),v=concat(v,n),)); v

More terms from Douglas Burke (dburke(AT)nevada.edu).

More terms from Hugo Pfoertner, Jun 23 2004

A046865 Numbers k such that 4*5^k - 1 is prime.

Original entry on oeis.org

0, 1, 3, 9, 13, 15, 25, 39, 69, 165, 171, 209, 339, 2033, 6583, 15393, 282989, 498483, 504221, 754611, 864751

Offset: 1

N. J. A. Sloane

a(22) > 1000000. - Karsten Bonath, Apr 04 2019

R. K. Guy, Unsolved Problems in Number Theory, Section A3.

Steven Harvey, Williams Primes
H. C. Williams, The primality of certain integers of the form 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.

Cf. A003307, A002235, A079906, A046866, A001771, A005541, A056725, A046867, A079907.

Print[0]; Do[ If[ PrimeQ[4*5^n - 1], Print[n]], {n, 1, 8100, 2}]

is(n)=isprime(4*5^n-1) \\ Charles R Greathouse IV, Feb 07 2017

Two more terms from Robert G. Wilson v, Jan 16 2003 and Jan 26 2003
a(16) from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 05 2008
a(1)=0 prepended by Robert Price, Feb 27 2015
a(17) from Karsten Bonath, Dec 07 2018
a(18)-a(19) from Karsten Bonath, Jan 17 2019
a(20)-a(21) from Karsten Bonath, Apr 04 2019

A046867 Numbers n such that 10*11^n -1 is prime.

Original entry on oeis.org

1, 3, 37, 119, 255, 355, 371, 497, 1759, 34863, 50719, 147709

Offset: 1

N. J. A. Sloane

a(13) > 2*10^5. - Robert Price, Jan 19 2015

R. K. Guy, Unsolved Problems in Theory of Numbers, Section A3.

H. C. Williams, The primality of certain integers of the form 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.

Cf. A003307, A002235, A046865, A079906, A046866, A001771, A005541, A056725, A079907.

Do[ If[ PrimeQ[10*11^n - 1], Print[n]], {n, 1, 2000}]

is(n)=isprime(10*11^n-1) \\ Charles R Greathouse IV, Feb 17 2017

One more term from Robert G. Wilson v, Jan 16 2003

a(10)-a(12) from Robert Price, Jan 19 2015

A079906 Numbers k such that 5*6^k - 1 is prime.

Original entry on oeis.org

1, 2, 6, 7, 11, 23, 33, 48, 68, 79, 116, 151, 205, 1016, 1332, 1448, 3481, 3566, 3665, 11233, 13363, 29166, 44358, 58530, 191706, 386450, 605168, 616879, 1204077

Offset: 1

Robert G. Wilson v, Jan 16 2003

a(29) > 618000. - Karsten Bonath, Nov 04 2019

R. K. Guy, Unsolved Problems in Theory of Numbers, Section A3.

H. C. Williams, The primality of certain integers of the form 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.

Cf. A003307, A002235, A046865, A046866, A001771, A005541, A056725, A046867, A079907, A247260.

Do[ If[ PrimeQ[5*6^n - 1], Print[n]], {n, 1, 5000}]

for(n=1,2000, if(isprime(5*6^n-1),print1(n, ", ")))

a(20)-a(24) from Donovan Johnson, Nov 26 2008
a(25) from Robert Price, Jan 23 2016
a(26) from Karsten Bonath, Jul 01 2019
a(27) from Karsten Bonath, Oct 29 2019
a(28) from Karsten Bonath, Nov 04 2019
a(29) from Ryan Propper, Nov 21 2023

A079907 Numbers n such that 11*12^n -1 is prime.

Original entry on oeis.org

1, 2, 21, 25, 33, 54, 78, 235, 1566, 2273, 2310, 4121, 7775, 42249, 105974, 138961

Offset: 1

Robert G. Wilson v, Jan 16 2003

a(17) > 2*10^5. - Robert Price, Mar 20 2015

R. K. Guy, Unsolved Problems in Theory of Numbers, Section A3.

Steven Harvey, Williams Primes
H. C. Williams, The primality of certain integers of the form 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.

Cf. A003307, A002235, A046865, A079906, A046866, A001771, A005541, A056725, A046867.

[n: n in [1..600]| IsPrime(11*12^n - 1)]; // Vincenzo Librandi, Mar 21 2015

Do[ If[ PrimeQ[11*12^n - 1], Print[n]], {n, 1, 2000}]
Select[Range[10000], PrimeQ[(11 12^# - 1)] &] (* Vincenzo Librandi, Mar 21 2015 *)

for(n=1,2000, if(isprime(11*12^n-1),print1(n, ", ")))

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 05 2008

a(13)-a(16) from Robert Price, Mar 20 2015

A046866 Numbers k such that 6*7^k - 1 is prime.

Original entry on oeis.org

0, 1, 2, 7, 18, 55, 69, 87, 119, 141, 189, 249, 354, 1586, 2135, 2865, 2930, 4214, 7167, 67485, 74402, 79326, 231349

Offset: 1

N. J. A. Sloane

a(23) > 2*10^5. - Robert Price, Nov 13 2015

R. K. Guy, Unsolved Problems in Number Theory, Section A3.

H. C. Williams, The primality of certain integers of the form 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.

Cf. A003307, A002235, A046865, A079906, A001771, A005541, A056725, A046867, A079907.

Do[ If[ PrimeQ[6*7^n - 1], Print[n]], {n, 0, 5650}]

for(n=0,2000, if(isprime(6*7^n-1),print1(n, ", ")))

One more term from Jason Earls, Jul 21 2001
More terms from Robert G. Wilson v, Jan 17 2003
One more term from Ryan Propper, Jun 05 2006
a(20)-a(22) from Donovan Johnson, Nov 26 2008
First term 0 inserted by Georg Fischer, Aug 01 2019
a(23) from Riley Fisher, Dec 02 2024

A005541 Numbers k such that 8*3^k - 1 is prime.

Original entry on oeis.org

0, 1, 2, 4, 10, 17, 50, 170, 184, 194, 209, 641, 1298, 4034, 5956, 7154, 9970, 35956, 42730, 132004, 190610

Offset: 1

N. J. A. Sloane

a(22) > 2*10^5. - Robert Price, Mar 16 2014
All terms are verified primes (i.e., not probable primes). - Robert Price, Mar 16 2014
896701 is a term, found in 2010 (see link). - Jeppe Stig Nielsen, Jul 31 2020

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

The Prime Pages, 8*3^896701 - 1.
H. C. Williams and C. R. Zarnke, Some prime numbers of the forms 2*3^n+1 and 2*3^n-1, Math. Comp., 26 (1972), 995-998.

Cf. A003307, A002235, A046865, A079906, A046866, A001771, A056725, A046867, A079907.

lst={};Do[If[PrimeQ[8*3^n-1], AppendTo[lst, n]], {n, 13^3}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 08 2008 *)

is(n)=isprime(8*3^n-1) \\ Charles R Greathouse IV, Feb 17 2017

More terms from Douglas Burke (dburke(AT)nevada.edu)
0 prepended by Vincenzo Librandi, Sep 26 2012
a(18)-a(21) from Robert Price, Mar 16 2014

A268061 Numbers k such that 7*8^k - 1 is prime.

Original entry on oeis.org

3, 7, 15, 59, 6127, 8703, 11619, 23403, 124299

Offset: 1

Robert Price, Jan 25 2016

a(10) > 2*10^5.

Terms are A001771(n)/3 that are integers.

R. K. Guy, Unsolved Problems in Theory of Numbers, Section A3.

H. C. Williams, The primality of certain integers of the form 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.

Cf. similar sequences of the form k*(k+1)^n-1: A003307 (k=2), ... (k=3), A046865 (k=4), A079906 (k=5), A046866 (k=6), this sequence (k=7), ... (k=8), A056725 (k=9), A046867 (k=10), A079907 (k=11).

Cf. A001771, A002235, A005541, A247260.

Select[Range[0, 200000], PrimeQ[7*8^# - 1] &]

lista(nn) = for(n=1, nn, if(ispseudoprime(7*8^n-1), print1(n, ", "))) \\ Altug Alkan, Jan 25 2016

cp's OEIS Frontend

A002235 Numbers m such that 3*2^m - 1 is prime.

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A003307 Numbers k such that 2*3^k - 1 is prime.

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A001771 Numbers k such that 7*2^k - 1 is prime.

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A046865 Numbers k such that 4*5^k - 1 is prime.

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A046867 Numbers n such that 10*11^n -1 is prime.

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A079906 Numbers k such that 5*6^k - 1 is prime.

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A079907 Numbers n such that 11*12^n -1 is prime.

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