A002957 -id:A002957 - cp's OEIS frontend

A055558 Primes of the form 1999...999.

19, 199, 1999, 199999, 19999999, 199999999999999999999999999, 1999999999999999999999999999, 199999999999999999999999999999999999999999999999999999

Offset: 1

Labos Elemer, Jul 10 2000

Primes of the form 2*10^k - 1.

			2*10^n - 1 is prime for {1,2,3,5,7,26,27,53,147,236,248,386,401}; in each of these numbers, the digit '9' appears n times.

Subsequence of A090149.
Primes in A067272.
Cf. A002957, A051200.

for(n=1,99,if(ispseudoprime(t=2*10^n-1),print1(t", "))) \\ Charles R Greathouse IV, Dec 10 2013

a(n) = 2*10^A002957(n) - 1 = A067272(A002957(n) + 1). - Elmo R. Oliveira, Jun 14 2025

A120378 Integers n such that 2*11^n-1 is prime.

Original entry on oeis.org

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

Offset: 1

Walter Kehowski, Jun 28 2006

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

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

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

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;

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

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

More terms from Ryan Propper, Jan 14 2008

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

A120375 Integers k such that 2*5^k - 1 is prime.

Original entry on oeis.org

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

Offset: 1

Walter Kehowski, Jun 28 2006

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

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

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.

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

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;

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

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

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

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

A120377 Primes of the form 2*11^k-1.

Original entry on oeis.org

241, 428717761

Offset: 1

Walter Kehowski, Jun 28 2006

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 values of k < 1000 that yield primes are 2, 8, 248. - T. D. Noe, Nov 16 2006

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

Amiram Eldar, Table of n, a(n) for n = 1..3

Cf. A000043, A000668, A002957, A002959, A003307, A055558, A079363, A120378.

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;

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

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

Corrected by T. D. Noe, Nov 16 2006

A119591 Least k such that 2*n^k - 1 is prime.

Original entry on oeis.org

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

Offset: 2

Pierre CAMI, Jun 01 2006

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)

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

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).

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

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

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)

Corrected and extended by Ray Chandler, Jun 08 2006

A171268 Primes q such that q^p ends with q, where p is the product of the digits of q.

Original entry on oeis.org

5, 11, 37, 61, 73, 199, 751, 3761, 7993, 79193, 7799999, 1111111111111111111, 11111111111111111111111, 199999999999999999999999999

Offset: 1

Farideh Firoozbakht, Apr 28 2010

All repunit primes (A004022) are in the sequence.
Number 2*10^k-1 is a term whenever k is an even term of A002957. - Max Alekseyev, Jun 08 2018
a(15) = 38*10^152-1, a(16) = 2*10^236-1, a(17) = 2*10^248-1, a(18) = (10^317-1)/9, a(19) = 38*10^352-1, a(20) = 2*10^386-1, a(21) = 78*10^535-1, a(22) = 2*10^546-1 are too large to include here. - Max Alekseyev, Jun 26 2018

			7799999^(7*7*9*9*9*9*9) == 7799999 (mod 10^7), so 7799999 is a term.

Max Alekseyev, Table of n, a(n) for n = 1..22

Cf. A004022, A171267, A171269.

Do[n=Prime[m];a=IntegerDigits[n];If[PowerMod[n,Apply[Times,a],10^Length[a]]==n,Print[n]],{m,100000000}]

a(12)-a(14) from Max Alekseyev, Aug 18 2013

A120376 Primes of the form 2*5^k - 1.

Original entry on oeis.org

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

Offset: 1

Walter Kehowski, Jun 28 2006

nonn

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

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

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.

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;

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

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

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

A171269 Intersection of A171267 and A171268.

Original entry on oeis.org

5, 751, 79193

Offset: 1

Farideh Firoozbakht, Apr 28 2010

Primes p such that both numbers p^A007953(p) and p^A007954(p) end with p.
a(4) > prime(10^8)=2038074743.
Number 2*10^k-1 is a term as soon as k is a multiple of 10 and belongs to A002957. In particular, a(4) <= 2*10^3020 - 1. - Max Alekseyev, Jun 07 2018

Cf. A171267, A171268.

A319535 Primes of the form 2*6^k - 1.

Original entry on oeis.org

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

Offset: 1

Jianing Song, Sep 22 2018

nonn

Primes in A164559.

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

			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.

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

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.

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

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

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

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

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

A129990 Primes p such that the smallest integer whose sum of decimal digits is p is prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 19, 23, 29, 31, 41, 43, 71, 79, 97, 173, 179, 257, 269, 311, 389, 691, 4957, 8423, 11801, 14621, 25621, 26951, 38993, 75743, 102031, 191671, 668869

Offset: 1

J. M. Bergot, Jun 14 2007

			The smallest integer whose sum of digits is 17 is 89; 89 is prime, therefore 17 is in the sequence.

Cf. A051885.

Cf. A002957, A056703, A056712, A056716, A056721, A056725.

Select[Prime[Range[1000]],PrimeQ[FromDigits[Join[{Mod[ #,9]},Table[9,{i,1,Floor[ #/9]}]]]] &]

from itertools import islice
from sympy import isprime, nextprime
def A051885(n): return ((n%9)+1)*10**(n//9)-1 # from Chai Wah Wu
def agen(startp=2):
    p = startp
    while True:
        if isprime(A051885(p)): yield p
        p = nextprime(p)
print(list(islice(agen(), 23))) # Michael S. Branicky, Jul 27 2022

sorted( filter(is_prime, sum(([9*t+k for t in oeis(seq).first_terms()] for seq,k in (('A002957',1), ('A056703',2), ('A056712',4), ('A056716',5), ('A056721',7), ('A056725',8))), [3])) ) # Max Alekseyev, Feb 05 2025

Primes p such that (p mod 9 + 1) * 10^[p/9] - 1 is prime. Therefore the sequence consists of the term 3 and the primes of the forms A002957(k)*9+1, A056703(k)*9+2, A056712(k)*9+4, A056716(k)*9+5, A056721(k)*9+7, A056725(k)*9+8. - Max Alekseyev, Nov 09 2009

Edited, corrected and extended by Stefan Steinerberger, Jun 23 2007
Extended by D. S. McNeil, Mar 20 2009
a(29)-a(33) from Max Alekseyev, Nov 09 2009

cp's OEIS Frontend

A055558 Primes of the form 1999...999.

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A120378 Integers n such that 2*11^n-1 is prime.

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Maple

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A120375 Integers k such that 2*5^k - 1 is prime.

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Magma

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A120377 Primes of the form 2*11^k-1.

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Maple

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A119591 Least k such that 2*n^k - 1 is prime.

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A171268 Primes q such that q^p ends with q, where p is the product of the digits of q.

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A120376 Primes of the form 2*5^k - 1.

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