A002959 -id:A002959

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A120378

Integers n such that 2*11^n-1 is prime.

+10
6

2, 8, 248, 2474, 2900, 6600, 24746, 105704

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See comments for A057472. Examined in base 12, all n must be even and all primes must be 1-primes. For example, 241 is 181 in base 12.

a(9) > 2*10^5. - Robert Price, Nov 06 2015

LINKS

Table of n, a(n) for n=1..8.

FORMULA

a(n) = n-th integer k such that 2*11^k-1 is prime.

EXAMPLE

a(1)=2 since 2*11^2-1=241 is the first prime of this form.

MAPLE

for w to 1 do for k from 1 to 2000 do n:=2*11^k-1; if isprime(n) then printf("%d, %d", k, n) fi od od;

MATHEMATICA

Select[Range[0, 200000], PrimeQ[2*11^# - 1] &] (* Robert Price, Nov 06 2015 *)

CROSSREFS

Cf. A000043, A000668, A002957, A002959, A003307, A079363, A055558.

KEYWORD

more,nonn

AUTHOR

Walter Kehowski, Jun 28 2006

EXTENSIONS

More terms from Ryan Propper, Jan 14 2008

a(7)-a(8) from Robert Price, Nov 06 2015

STATUS

approved

A120375

Integers k such that 2*5^k - 1 is prime.

+10
5

4, 6, 16, 24, 30, 54, 96, 178, 274, 1332, 2766, 3060, 4204, 17736, 190062, 223536, 260400, 683080

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See comments for A057472. Examined in base 12, all n must be even and all primes must be 1-primes. For example, 1249 is 881 in base 12.

a(16) > 2*10^5. - Robert Price, Mar 14 2015

LINKS

Table of n, a(n) for n=1..18.

FORMULA

a(n) = 2*A002958(n).

EXAMPLE

a(1) = 4 since 2*5^4 - 1 = 1249 is the first prime.

MAPLE

for w to 1 do for k from 1 to 2000 do n:=2*5^k-1; if isprime(n) then printf("%d, %d ", k, n) fi od od;

MATHEMATICA

Select[Range[0, 100], PrimeQ[2*5^# - 1] &] (* Robert Price, Mar 14 2015 *)

PROG

(PARI) isok(k) = ispseudoprime(2*5^k-1); \\ Altug Alkan, Sep 22 2018

(Magma) [n: n in [0..2800] |IsPrime(2*5^n - 1)]; // Vincenzo Librandi, Sep 23 2018

CROSSREFS

Integers k such that 2*b^k - 1 is prime: A090748 (b=2), A003307 (b=3), this sequence (b=5), A057472 (b=6), A002959 (b=7), A002957 (b=10), A120378 (b=11).

Primes of the form 2*b^k - 1: A000668 (b=2), A079363 (b=3), A120376 (b=5), A158795 (b=7), A055558 (b=10), A120377 (b=11).

Cf. also A000043, A002958.

KEYWORD

nonn,more

AUTHOR

Walter Kehowski, Jun 28 2006

EXTENSIONS

More terms from Ryan Propper, Mar 28 2007

a(14) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 02 2007

a(15) from Robert Price, Mar 14 2015

a(16)-a(18) from Jorge Coveiro and Tyler NeSmith, Jun 14 2020

STATUS

approved

A158795

Primes of the form 2*7^k - 1.

+10
5

13, 97, 4801, 33613, 1356446145697, 383162462761132828801, 6439811511626359453675213, 5303461691719306943558046763201, 1498096661930372466988988205389128987297, 81072431194288773664131732218033347601750444502024167492384908896001

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Values of k are in A002959.

All terms are of the form 12*k + 1. - Altug Alkan, Sep 22 2018

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..12

EXAMPLE

2*7^1 - 1 = 13, 2*7^2 - 1 = 97, and 2*7^4 - 1 = 4801 are primes, but 2*7^3 - 1 = 685 is not.

MATHEMATICA

Select[Table[2*7^n-1, {n, 0, 300}], PrimeQ] (* Vincenzo Librandi, Jul 26 2012 *)

PROG

(Magma) [a: n in [0..100] | IsPrime(a) where a is 2*7^n-1 ]; // Vincenzo Librandi, Jul 26 2012

(PARI) for(k=1, 1e3, if(ispseudoprime(p=2*7^k-1), print1(p, ", "))); \\ Altug Alkan, Sep 22 2018

CROSSREFS

Cf. A002959.

KEYWORD

nonn

AUTHOR

Vincenzo Librandi, Mar 27 2009

EXTENSIONS

a(7) corrected and examples edited by Jon E. Schoenfield, Jun 19 2010

STATUS

approved

A119591

Least k such that 2*n^k - 1 is prime.

+10
3

1, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 2, 4, 1, 1, 2, 2, 1, 10, 1, 1, 6, 1, 2, 6, 1, 2, 136, 1, 1, 6, 6, 1, 6, 1, 1, 2, 2, 1, 2, 1, 2, 4, 1, 2, 4, 4, 1, 2, 1, 1, 44, 1, 1, 2, 1, 3, 2, 5, 3, 2, 2, 1, 4, 1, 768, 4, 1, 1, 52, 34, 2, 132, 1, 1, 14, 7, 1, 2, 2, 1, 8, 1, 2, 10, 1, 24, 60, 1, 1, 2, 3, 5, 2, 1, 1, 2, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,4

COMMENTS

From Eric Chen, Jun 01 2015: (Start)

Conjecture: a(n) is defined for all n.

a(303) > 10000, a(304)..a(360) = {1, 2, 11, 1, 990, 1, 1, 2, 2, 4, 74, 5, 1, 10, 6, 6, 4, 1, 1, 2, 1, 9, 12, 1, 80, 2, 1, 1, 2, 14, 3, 2, 3, 1, 12, 1, 60, 36, 1, 8, 4, 34, 1, 522, 3, 15, 14, 1, 6, 2, 3, 1, 4, 5, 4, 10, 1}.

a(n) = 1 if and only if n is in A006254. (End)

From Eric Chen, Sep 16 2021: (Start)

Now a(303) is known to be 40174, also other terms > 10000: a(383) = 20956, a(515) = 58466, a(522) = 62288, a(578) = 129468, a(581) > 400000, a(590) = 15526, a(647) = 21576, a(662) = 16590, a(698) = 127558, a(704) = 62034, see the a-file and the references.

a(n) = 2 if and only if n is in A066049 but not in A006254.

a(n) = 3 if and only if n is in A214289 but not in A006254 or A066049. (End)

LINKS

Eric Chen, Table of n, a(n) for n = 2..580

Gary Barnes, Riesel conjectures and proofs

Eric Chen, Table of n, a(n) for n = 2..2050 status

Prime Wiki, Riesel prime small bases least n

FORMULA

From Eric Chen, Sep 16 2021: (Start)

a(6*n) = A098873(n).

a(2^n) = A279095(n).

a(A006254(n)) = 1.

a(A066049(n)) <= 2.

a(A214289(n)) <= 3. (End)

MATHEMATICA

f[n_] := Block[{k = 0}, While[ ! PrimeQ[2*n^k - 1], k++ ]; k ]; Table[f[n], {n, 2, 106}] (* Ray Chandler, Jun 08 2006 *)

PROG

(PARI) a(n) = for(k=1, 2^24, if(ispseudoprime(2*n^k-1), return(k))) \\ Eric Chen, Jun 01 2015

CROSSREFS

Cf. A119624, A253178.

Numbers r such that 2*k^r-1 is prime: A090748 (k=2), A003307 (k=3), A146768 (k=4), A120375 (k=5), A057472 (k=6), A002959 (k=7), ... (k=8), ... (k=9), A002957 (k=10), A120378 (k=11), ... (k=12), A174153 (k=13), A273517 (k=14), ... (k=15), ... (k=16), A193177 (k=17), A002958 (k=25).

KEYWORD

nonn,hard

AUTHOR

Pierre CAMI, Jun 01 2006

EXTENSIONS

Corrected and extended by Ray Chandler, Jun 08 2006

STATUS

approved

A120377

Primes of the form 2*11^n-1.

+10
3

241, 428717761

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See comments for A057472. Examined in base 12, all n must be even and all primes must be 1-primes. For example, 241 is 181 in base 12.

The n<1000 that yield primes are 2, 8, 248. - T. D. Noe, Nov 16 2006

LINKS

Table of n, a(n) for n=1..2.

FORMULA

a(n) = n-th number such that 2*11^k-1 that is prime for some k.

a(n) = 2*11^A120378(n)-1. - R. J. Mathar, Mar 06 2010

EXAMPLE

a(1)=241 since 2*11^2-1=241 is the first prime.

MAPLE

for w to 1 do for k from 1 to 2000 do n:=2*11^k-1; if isprime(n) then printf("%d, %d", k, n) fi od od;

MATHEMATICA

Select[2*11^Range[1000]-1, PrimeQ] (* T. D. Noe, Nov 16 2006 *)

CROSSREFS

Cf. A000043, A000668, A002957, A002959, A003307, A079363, A055558.

KEYWORD

nonn,bref

AUTHOR

Walter Kehowski, Jun 28 2006

EXTENSIONS

Corrected by T. D. Noe, Nov 16 2006

STATUS

approved

A120376

Primes of the form 2*5^k - 1.

+10
2

1249, 31249, 305175781249, 119209289550781249, 1862645149230957031249, 111022302462515654042363166809082031249, 25243548967072377773175314089049159349542605923488736152648925781249

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See comments for A057472. Examined in base 12, all n must be even and all primes must be 1-primes. For example, 1249 is 881 in base 12.

The next term has 125 digits. - Harvey P. Dale, Jan 26 2019

LINKS

Table of n, a(n) for n=1..7.

FORMULA

a(n) = 2*5^A120375(n) - 1 = 2*5^(2*A002958(n)) - 1. - Jianing Song, Sep 22 2018

EXAMPLE

a(1) = 4 since 2*5^4 - 1 = 1249 is the first prime.

MAPLE

for w to 1 do for k from 1 to 2000 do n:=2*5^k-1; if isprime(n) then printf("%d, %d", k, n) fi od od;

MATHEMATICA

Select[2*5^Range[100]-1, PrimeQ] (* Harvey P. Dale, Jan 26 2019 *)

PROG

(PARI) for(k=1, 1e3, if(ispseudoprime(p=2*5^k-1), print1(p, ", "))); \\ Altug Alkan, Sep 22 2018

CROSSREFS

Integers k such that 2*b^k - 1 is prime: A090748 (b=2), A003307 (b=3), A120375 (b=5), A057472 (b=6), A002959 (b=7), A002957 (b=10), A120378 (b=11).

Primes of the form 2*b^k - 1: A000668 (b=2), A079363 (b=3), this sequence (b=5), A158795 (b=7), A055558 (b=10), A120377 (b=11).

Cf. also A000043, A002958.

KEYWORD

nonn

AUTHOR

Walter Kehowski, Jun 28 2006

STATUS

approved

A319535

Primes of the form 2*6^k - 1.

+10
1

11, 71, 431, 2591, 15551, 4353564671, 5642219814911, 341163456359156416511, 2046980738154938499071, 20628849596981071092343898111, 26734989077687468135677691953151, 207891275068097752223029732627709951, 269427092488254686881046533485512097791

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Primes in A164559.

Companion sequence of A057472. There are 49 terms known in this sequence.

LINKS

K. D. Bajpai, Table of n, a(n) for n = 1..26

FORMULA

a(n) = 2*6^A057472(n) - 1.

EXAMPLE

2*6^1 - 1 = 11, 2*6^2 - 1 = 71, 2*6^3 - 1 = 431, 2*6^4 - 1 = 2591 and 2*6^5 - 1 = 15551 are primes, but 2*6^6 - 1 = 93311 = 23*4057 is not.

MAPLE

A319535:= n-> (2*6^n-1): select(isprime, [seq((A319535(n), n=1..200))]); # K. D. Bajpai, Nov 15 2019

MATHEMATICA

Select[Table[2*6^k-1, {k, 1600}], PrimeQ[#]&] (* K. D. Bajpai, Nov 15 2019 *)

PROG

(PARI) for(n=1, 99, my(t); if(ispseudoprime(t=2*6^n-1), print1(t", ")))

(Magma) [k: n in [1..100] | IsPrime(k) where k is 2*6^n-1]; // K. D. Bajpai, Nov 15 2019

CROSSREFS

Integers k such that 2*b^k - 1 is prime: A090748 (b=2), A003307 (b=3), A120375 (b=5), A057472 (b=6), A002959 (b=7), A002957 (b=10), A120378 (b=11).

Primes of the form 2*b^k - 1: A000668 (b=2), A079363 (b=3), A120376 (b=5), this sequence (b=6), A158795 (b=7), A055558 (b=10), A120377 (b=11).

Cf. also A000043, A002958, A068231, A164559.

KEYWORD

nonn

AUTHOR

Jianing Song, Sep 22 2018

STATUS

approved

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