A055558 -id:A055558 - cp's OEIS frontend

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

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

A141311 Primes consisting of a digit and a nonempty string of 9's (i.e., primes of the form k*10^m - 1, where k is any digit).

Original entry on oeis.org

19, 29, 59, 79, 89, 199, 499, 599, 1999, 2999, 4999, 8999, 49999, 59999, 79999, 199999, 599999, 799999, 2999999, 4999999, 19999999, 29999999, 59999999, 89999999, 799999999, 59999999999, 79999999999, 59999999999999, 499999999999999, 29999999999999999999

Offset: 1

Lekraj Beedassy, Aug 02 2008

k can never be 1, 4, or 7, because if it were, k*10^m - 1 would be divisible by 3.

Jon E. Schoenfield, Table of n, a(n) for n = 1..71 (next term is 8*10^1219 - 1)

Cf. A055558, A065582.

d={9};s={};Do[Do[m=FromDigits[Join[IntegerDigits[i],d]];If[PrimeQ[m],AppendTo[s,m]],{i,8}];AppendTo[d,9],{j,19}];s (* James C. McMahon, Jul 20 2025 *)

59999999 from Howard Berman (howard_berman(AT)hotmail.com), Apr 22 2009

Two more terms from Jon E. Schoenfield, Jan 12 2019

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

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

A320256 k-digit primes with the same even digit repeated k-1 times and a single odd digit.

Original entry on oeis.org

3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 223, 227, 229, 443, 449, 661, 881, 883, 887, 2221, 4441, 4447, 6661, 8887, 22229, 44449, 88883, 444443, 444449, 666667, 888887, 22222223, 66666667, 88888883, 222222227, 444444443, 666666667, 888888883, 888888887

Offset: 1

Enrique Navarrete, Oct 08 2018

For the resulting number to be prime, the rightmost digit must be the odd one. - Michel Marcus, Oct 11 2018

			3, 5, 7 are in the sequence for k = 1.
229 is in the sequence because it is a 3-digit prime with the first 3-1 digits repeating even (2) and the last digit odd (9). - _David A. Corneth_, Oct 10 2018

Alois P. Heinz, Table of n, a(n) for n = 1..132

Cf. A055558, A068690, A105975, A141311, A154764.

Join[{3, 5, 7}, Select[Flatten@ Table[{1, 3, 7, 9} + 10 FromDigits@ ConstantArray[k, n], {n, 9}, {k, Range[2, 8, 2]}], PrimeQ]] (* Michael De Vlieger, Oct 31 2018 *)

first(n) = {n = max(n, 3); my(t = 3, res = List([3, 5, 7])); print1("3, 5, 7, "); for(i=1, oo, k=(10^i - 1) / 9; forstep(f = 2, 8, 2, forstep(d=1, 9, 2, c = 10 * f * k + d; if(isprime(c), print1(c", "); listput(res, c); t++; if(t>=n, return(res))))))} \\ David A. Corneth, Oct 10 2018

More terms from Michel Marcus, Oct 10 2018

A164890 Primes composed of digit {1,9} and with digit sum 9*k+1.

Original entry on oeis.org

19, 199, 919, 991, 1999, 9199, 99991, 199999, 991999, 999199, 9999991, 19999999, 99991999, 9199999999, 11111111911, 11119111111, 99999199999, 99999991999, 111111911191, 111191119111, 111911191111, 191119111111, 991999999999, 999999991999, 1111111119919, 1111111191199, 1111111191919, 1111111199119

Offset: 1

Zak Seidov, Aug 29 2009

Corresponding k's are 1, 2, 2, 2, 3, 3, 4, 5, 5, 5, 6, 7, 7, 9, 2, 2, 10, 10, 3, 3, 3, 3, 11, 11, 4, 4, 4, 4. - Robert Israel, May 02 2018

Number of primes having a digital length of k=1,2,3...: 0, 1, 3, 2, 1, 3, 1, 2, 0, 1, 4, 6, 33, 81, 329, 455, 2028, 3134, 9193, 9060, 31615, 39246, 88069, 94794, 252965, 309437, ..., . = Robert G. Wilson v, May 05 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007953, A055558.

Res:= {}:
for d from 2 to 14 do
  for j from 1 to d by 9 do
    Res:= Res union select(isprime, {seq((10^d-1)/9 + 8*add(10^i,i=s), s = combinat:-choose([$0..d-1],d-j))})
od od:
sort(convert(Res,list)); # Robert Israel, May 02 2018

f[n_] := Block[{s, t = Tuples[{1, 9}, n]}, s = Select[t, Mod[Plus @@ #, 9] == 1 &]; Select[ FromDigits@# & /@ s, PrimeQ]]; Array[f, 12] // Flatten (* Robert G. Wilson v, May 04 2018 *)

isok(n) = isprime(n) && (Set(digits(n)) == [1, 9]) && ((sumdigits(n) % 9) == 1); \\ Michel Marcus, Oct 16 2013

Definition corrected by Michel Marcus, Oct 16 2013

Corrected by Robert Israel, May 02 2018

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.

A378950 Numbers that are a proper substring of the concatenation (with repetition) in decreasing order of their prime factors.

Original entry on oeis.org

95, 132, 995, 9995, 73332, 85713, 93115, 131131, 197591, 632812, 999995, 4285713, 8691315, 58730137, 99999995, 131373333, 507107133, 4870313015

Offset: 1

Scott R. Shannon, Dec 11 2024

All numbers of the form 5*A055558(k), k>=1, are terms.

			95 is a term as 95 = 19 * 5 = "195" when concatenated, which contains "95" as a substring.
632812 is a term as 632812 = 563 * 281 * 2 * 2 = "56328122" when concatenated, which contains "632812" as a substring.
4870313015 is a term as 4870313015 = 748703 * 1301 * 5 = "74870313015" when concatenated, which contains "4870313015" as a substring.

Cf. A027746, A378893, A378894, A376078, A372309.

A160452 Invertible primes of the form 1 followed by a string of 9's.

Original entry on oeis.org

19, 199, 1999, 1999999999999999999999999999

Offset: 1

Lekraj Beedassy, May 14 2009

These are values in A055558 whose rotation by 180 degrees occurs in A092571. I have checked all the numbers that correspond to entries in A002957 and can confirm that the next term in this sequence, if it exists, is greater than 2*10^55347-1. [From Dmitry Kamenetsky, May 22 2009]

			1999, for instance, is a prime which rotated upside down through 180 degrees becomes the prime 6661. Hence 1999 is in the sequence.

Cf. A048890, A055558.

Cf. A092571, A002957. [From Dmitry Kamenetsky, May 22 2009]

cp's OEIS Frontend

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

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

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A141311 Primes consisting of a digit and a nonempty string of 9's (i.e., primes of the form k*10^m - 1, where k is any digit).

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

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

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A320256 k-digit primes with the same even digit repeated k-1 times and a single odd digit.

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A164890 Primes composed of digit {1,9} and with digit sum 9*k+1.

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