A056805 -id:A056805 - cp's OEIS frontend

A056716 Numbers n such that 6*10^n-1 is prime.

0, 1, 2, 4, 5, 7, 10, 13, 22, 23, 28, 34, 40, 61, 73, 361, 490, 613, 1624, 2000, 2994, 4301, 4332, 18668, 32544, 34936, 267598, 270658, 293134, 319889, 414508

Offset: 1

Robert G. Wilson v, Aug 11 2000

Also numbers n such that 5*10^n + 9*R_n is prime, where R_n = 11...1 is the repunit (A002275) of length n.

Makoto Kamada, Prime numbers of the form 599...99.
Index entries for primes involving repunits

Cf. A056805 (6*10^n+1 is prime).

Do[ If[ PrimeQ[ 6*10^n - 1], Print[n]], {n, 0, 5000}]

is(n)=ispseudoprime(6*10^n-1) \\ Charles R Greathouse IV, Feb 20 2017

More terms from Hugo Pfoertner, Feb 12 2004
3 more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 30 2007
a(27)-a(31) from Kamada data by Robert Price, Oct 19 2014
Outdated comment deleted by Harvey P. Dale, Mar 01 2020

A101517 Indices of primes in sequence defined by A(0) = 61, A(n) = 10*A(n-1) - 9 for n > 0.

Original entry on oeis.org

0, 1, 7, 8, 14, 19, 25, 37, 44, 64, 111, 243, 302, 392, 559, 838, 1008, 1018, 1172, 1333, 2235, 2628, 4425, 8847, 20811, 37743, 72925, 86286

Offset: 1

Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Dec 06 2004

Numbers n such that 60*10^n + 1 is prime.
Numbers n such that digit 6 followed by n >= 0 occurrences of digit 0 followed by digit 1 is prime.
Numbers corresponding to terms <= 838 are certified primes.
Certified primality of numbers corresponding to terms 1008,1018,1172,1333 with Primo. - Ryan Propper, Jun 20 2005

			600000001 is prime, hence 7 is a term.

Klaus Brockhaus and Walter Oberschelp, Zahlenfolgen mit homogenem Ziffernkern, MNU 59/8 (2006), pp. 462-467.

Makoto Kamada, Prime numbers of the form 600...001.

Cf. A000533, A002275, A056805.

a=61;for(n=0,1500,if(isprime(a),print1(n,","));a=10*a-9)

for(n=0,1500,if(isprime(60*10^n+1),print1(n,",")))

a(n) = A056805(n+1) - 1.

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm) and Stefan Steinerberger, Apr 28 2007
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 01 2008
a(26)-a(28) from Kamada data by Ray Chandler, Apr 23 2015

A109713 Numbers n such that 99 * 10^n + 1 is prime.

Original entry on oeis.org

1, 2, 4, 8, 16, 20, 24, 72, 200, 359, 454, 624, 1054, 2060, 6301, 8083, 8407, 13159, 65059, 74957

Offset: 1

Jason Earls, Aug 08 2005

Terms < 21000 have been certified. Primality proof for 13159: PFGW Version 20041001.Win_Stable (v1.2 RC1b) [FFT v23.8] Primality testing 99*10^13159+1 [N-1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 17 Calling Brillhart-Lehmer-Selfridge with factored part 69.89% 99*10^13159+1 is prime! (29.5646s+0.0028s) [Comment edited by N. J. A. Sloane, Jan 28 2025]

			For n=8 we have 99*10^8+1 = 9900000001, which is prime.

Makoto Kamada, Search for 990w1.

Cf. A056797, A056804, A056805, A056806, A056807.

is(n)=ispseudoprime(99*10^n+1) \\ Charles R Greathouse IV, Jun 12 2017

Edited by N. J. A. Sloane at the suggestion of Herman Jamke, Jan 13 2008

a(19)-a(20) from Kamada data by Tyler Busby, Apr 16 2024

A216889 Numbers k such that 12*3^k + 1 is prime.

Original entry on oeis.org

0, 1, 2, 5, 13, 14, 38, 200, 248, 884, 1004, 1253, 1634, 3305, 3521, 9601, 19784, 72697

Offset: 1

Vincenzo Librandi, Sep 26 2012

a(19) > 2*10^5. - Robert Price, Mar 16 2014

All terms are verified primes (i.e., not merely probable primes). - Robert Price, Mar 16 2014

Cf. A003306, A005537-A005539, A056805.

[n: n in [0..4000] | IsPrime(12*3^n + 1)];

Select[Range[4000], PrimeQ[12 * 3^# + 1] &]

is(n)=ispseudoprime(12*3^n+1) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = A005537(n+1) - 1. - Bruno Berselli, Sep 27 2012

a(16)-a(17) from Vincenzo Librandi, Sep 30 2012

a(18) from Robert Price, Mar 16 2014

A216890 Numbers n such that 14*3^n + 1 is prime.

Original entry on oeis.org

1, 2, 3, 18, 22, 26, 27, 33, 39, 57, 62, 94, 145, 246, 390, 398, 402, 571, 690, 906, 1062, 1254, 1367, 1627, 1954, 2409, 3107, 14754, 15378, 24219, 46138, 98883, 161178

Offset: 1

Vincenzo Librandi, Sep 26 2012

nonn

The next terms are > 6000.
a(34) > 2*10^5. - Robert Price, Mar 16 2014
All terms are verified primes (i.e., not probable primes). - Robert Price, Mar 16 2014

Cf. A003306, A005537-A005539, A056805.

[n: n in [0..4000] | IsPrime(14*3^n+1)];

Select[Range[4000], PrimeQ[14 3^# + 1] &]

is(n)=ispseudoprime(14*3^n+1) \\ Charles R Greathouse IV, Jun 13 2017

a(28)-a(33) from Robert Price, Mar 16 2014

A109800 Numbers n such that 55*10^n + 1 is prime.

Original entry on oeis.org

2, 3, 7, 9, 33, 61, 93, 112, 284, 615, 1293, 2558, 2925, 5961, 6454, 7960, 17521, 40838

Offset: 1

Jason Earls, Aug 15 2005

All values proved prime. No more up to 25000. Primality proof for the largest: PFGW Version 20041001.Win_Stable (v1.2 RC1b) [FFT v23.8] Primality testing 55*10^17521+1 [N-1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 3 Calling Brillhart-Lehmer-Selfridge with factored part 69.89% 55*10^17521+1 is prime! (48.1892s+0.1481s)

Cf. A056806, A056805.

is(n)=ispseudoprime(55*10^n+1) \\ Charles R Greathouse IV, Jun 13 2017

a(18) from Kamada data by Tyler Busby, Apr 24 2024

A171612 Integers n such that (25*10^n)+1 is prime.

Original entry on oeis.org

1, 8, 255, 320, 609, 688, 1436, 3271, 3921, 6520, 19604, 38348, 63531

Offset: 1

Julien Peter Benney (jpbenney(AT)ftml.net), Dec 13 2009

No others less than 20000.

See Kamada link - primecount.txt for terms, primesize.txt for discovery details including probable or proved primes - search on "25001".

			For n=8 we have (25*10^8)+1 = 25*100000000+1 = 2500000000+1 = 2500000001, which is prime.

Makoto Kamada, List of near-repdigit-related prime numbers.
Index entries for primes involving repunits.

Cf. A056797, A056804, A056805, A056806, A056807

Edited by Ray Chandler, Dec 23 2010

a(12)-a(13) from Kamada data by Tyler Busby, May 03 2024

A177132 Primes of the form 6*10^n+1.

Original entry on oeis.org

7, 61, 601, 600000001, 6000000001, 6000000000000001, 600000000000000000001, 600000000000000000000000001, 600000000000000000000000000000000000001, 6000000000000000000000000000000000000000000001, 600000000000000000000000000000000000000000000000000000000000000001

Offset: 1

Vincenzo Librandi, Dec 10 2010

nonn

Makoto Kamada, Prime numbers of the form 600...001.

Cf. A056805.

[a: n in [0..250]|IsPrime(a) where a is 6*10^n+1];

Select[Table[6*10^n+1,{n,0,70}],PrimeQ] (* Harvey P. Dale, Feb 22 2015 *)

A216888 Numbers k such that 6*3^k + 1 is prime.

Original entry on oeis.org

0, 1, 3, 4, 5, 8, 15, 16, 29, 53, 56, 59, 64, 131, 179, 319, 695, 781, 821, 896, 1251, 1453, 4216, 5479, 6224, 7841, 12095, 13781, 17719, 43955, 64821, 82779, 105105, 152528, 165895, 191813, 529679, 1074725, 1086111, 1175231, 1277861, 1346541, 3123035, 3648968, 5570080, 6236771, 10852676

Offset: 1

Vincenzo Librandi, Sep 26 2012

			3 is a term because 6*3^3 + 1 = 163 is prime.
7 is not a term because 6*3^7 + 1 = 13123 = 11*1193 is composite.

Cf. A003306, A005537, A005538, A005539; A056800, A056805, A143279.

Associated primes are in A111974.

/* Gives only the terms up to 1453: */ [n: n in [0..1500] | IsPrime(6*3^n + 1)];

Select[Range[5000], PrimeQ[6 3^# + 1] &]

is(n)=ispseudoprime(6*3^n+1) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = A003306(n+1)-1. - Bruno Berselli, Sep 27 2012

More terms from Vincenzo Librandi, Oct 01 2012

a(41)-a(47) from the data at A003306 added by Amiram Eldar, Jul 18 2025

A294396 Numbers k such that 12*10^k + 1 is prime.

Original entry on oeis.org

0, 2, 38, 80, 9230, 25598, 39500

Offset: 1

Patrick A. Thomas, Feb 12 2018

k must be even since 12*10^k + 1 is divisible by 11 if k is odd. - Robert G. Wilson v, Feb 12 2018
a(7) > 27440. - Robert G. Wilson v, Feb 17 2018
a(8) > 10^5. - Jeppe Stig Nielsen, Jan 31 2023

			13 and 1201 are prime, so 0 and 2 are the initial values.

Cf. A001562, A096507, A056797, A056807, A056806, A102940, A056805, A056804, A096508, A102975, A289051, A099017, A295325, A273002, A102945, A282456, A267420.

Cf. A062339 (primes with digit sum 4).

ParallelMap[ If[ PrimeQ[12*10^# +1], #, Nothing] &, 2 + 6Range@ 4500] (* Robert G. Wilson v, Feb 13 2018 *)

isok(k) = isprime(12*10^k + 1); \\ Altug Alkan, Mar 04 2018

a(5) from Robert G. Wilson v, Feb 12 2018
a(6) from Robert G. Wilson v, Feb 13 2018
a(7) from Jeppe Stig Nielsen, Jan 28 2023

cp's OEIS Frontend

A056716 Numbers n such that 6*10^n-1 is prime.

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A101517 Indices of primes in sequence defined by A(0) = 61, A(n) = 10*A(n-1) - 9 for n > 0.

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A109713 Numbers n such that 99 * 10^n + 1 is prime.

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A216889 Numbers k such that 12*3^k + 1 is prime.

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A216890 Numbers n such that 14*3^n + 1 is prime.

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A109800 Numbers n such that 55*10^n + 1 is prime.

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A171612 Integers n such that (25*10^n)+1 is prime.

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A177132 Primes of the form 6*10^n+1.

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