A056806 -id:A056806 - cp's OEIS frontend

A056797 Numbers k such that 9*10^k+1 is prime.

3, 4, 5, 9, 22, 27, 36, 57, 62, 78, 201, 537, 696, 790, 905, 1038, 66886, 70500, 91836, 100613, 127240, 380734, 583696, 719055, 823037, 862868

Offset: 1

Robert G. Wilson v, Aug 22 2005

a(22) > 2*10^5. - Robert Price, Jan 21 2015

			For k=9 we have (9*(10^9))+1 = 9000000001, which is prime.

Makoto Kamada, Prime numbers of the form 900...001.
Sabin Tabirca and Kieran Reynolds, Lacunary Prime Numbers.
Index entries for primes involving repunits.

Cf. A056806 (4*10^k+1 is prime), A100997.

[n: n in [0..1000] | IsPrime(9*10^n+1)]; // Vincenzo Librandi, May 25 2015

Do[ If[ PrimeQ[9*10^n + 1], Print[n]], {n, 0, 10000}]

is(n)=ispseudoprime(9*10^n+1) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = A100997(n) + 1.

a(18)-a(19) from Kamada data by Robert Price, Dec 14 2010
a(20) from Predrag Kurtovic, Sep 23 2013
a(21) from Robert Price, Jan 21 2015
a(22)-a(23) from Kamada data by Mohammed Yaseen, Jul 20 2021
a(24) from Predrag Kurtovic, Apr 18 2024
a(25)-a(26) from Predrag Kurtovic, Apr 22 2024

A101397 Numbers k such that 4*10^k+3 is prime.

Original entry on oeis.org

0, 1, 3, 7, 10, 40, 419, 449, 1737, 2245, 3131, 3813, 5345, 5659, 5681, 8410, 9097, 11293, 21061

Offset: 1

Julien Peter Benney (jpbenney(AT)ftml.net), Jan 15 2005

nonn

See Kamada link for search limit and prime vs. PRP status.

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

			n = 1, 3, 7, 10 are members since 43, 4003, 40000003 and 40000000003 are prime numbers.

Makoto Kamada, Prime numbers of the form 400...003.
Sabin Tabirca and Kieran Reynolds, Lacunary Prime Numbers.

Cf. A049054, A081677, A056806, A101713.

Do[ If[ PrimeQ[4*10^n + 3], Print[n]], {n, 0, 10000}] (* Robert G. Wilson v, Jan 18 2005 *)

is(n)=ispseudoprime(4*10^n+3) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = A101713(n-1) + 1.

a(18)-a(19) from Kamada data by Robert Price, Dec 10 2010

A177506 Primes of the form 4*10^n+1.

Original entry on oeis.org

5, 41, 401, 4001, 40000000000001

Offset: 1

Vincenzo Librandi, Dec 11 2010

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..9
Makoto Kamada, Prime numbers of the form 400...001.

Cf. A056806.

[a: n in [0..250] | IsPrime(a) where a is 4*10^n+1];

Select[Table[4 10^n + 1, {n, 0, 500}], PrimeQ] (* Vincenzo Librandi Jan 02 2014 *)

A101394 Numbers k such that 4*10^k+9 is prime.

Original entry on oeis.org

0, 2, 4, 5, 8, 9, 28, 191, 196, 2038, 34414, 39266, 50579, 94286, 108412, 130480, 178091, 185355

Offset: 1

Julien Peter Benney (jpbenney(AT)ftml.net), Jan 15 2005

nonn

a(19) > 2*10^5. - Robert Price, May 24 2015

			n = 2, 4, 5, 8, 9 are members since 409, 40009, 400009, 400000009 and 4000000009 are all prime.

Makoto Kamada, Prime numbers of the form 400...009.
Sabin Tabirca and Kieran Reynolds, Lacunary Prime Numbers.

Cf. A088275, A056806, A101715.

Do[ If[ PrimeQ[4*10^n + 9], Print[n]], {n, 0, 10000}]

is(n)=ispseudoprime(4*10^n+9) \\ Charles R Greathouse IV, Jun 12 2017

a(n) = A101715(n-1) + 1.

a(10)=2038 from Joao da Silva (zxawyh66(AT)yahoo.com), Sep 30 2005
a(11)-a(12) from Kamada data by Robert Price, Dec 13 2010
Edited by Ray Chandler, Dec 23 2010
a(13)-a(18) from Robert Price, May 24 2015

A101395 Numbers k such that 4*10^k+7 is prime.

Original entry on oeis.org

0, 1, 3, 9, 39, 2323

Offset: 1

Julien Peter Benney (jpbenney(AT)ftml.net), Jan 15 2005

No further terms < 50000.

a(7) > 2*10^5. - Robert Price May 16 2015

			n = 1, 3, 9 are members since 47, 4007 and 4000000007 are primes.

Makoto Kamada, Prime numbers of the form 400...007.
Sabin Tabirca and Kieran Reynolds, Lacunary Prime Numbers.
Index entries for primes involving repunits.

Cf. A056806, A100501, A088274, A101714.

Do[ If[ PrimeQ[4*10^n + 7], Print[n]], {n, 0, 10000}]

is(n)=ispseudoprime(4*10^n+7) \\ Charles R Greathouse IV, Jun 12 2017

a(n) = A101714(n-1) + 1.

A101712 Indices of primes in sequence defined by A(0) = 41, A(n) = 10*A(n-1) - 9 for n > 0.

Original entry on oeis.org

0, 1, 2, 12, 228, 241, 308, 956, 1472, 1493, 3181, 3726, 4176, 23209, 25718, 32834, 36989, 103957

Offset: 1

Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Dec 14 2004

Numbers n such that 40*10^n + 1 is prime.

Numbers n such that digit 4 followed by n >= 0 occurrences of digit 0 followed by digit 1 is prime.

			4001 is prime, hence 2 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 400...001.

Cf. A000533, A002275, A056806.

Select[Range[0, 1500], PrimeQ[4*10^# + 1] &] (* Robert Price, Mar 19 2015 *)

a=41;for(n=0,1500,if(isprime(a),print1(n,","));a=10*a-9)

for(n=0,1500,if(isprime(40*10^n+1),print1(n,",")))

a(n) = A056806(n+1) - 1.

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 02 2008

Added missing term a(16)=32834 by Robert Price, Mar 19 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

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

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

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A056797 Numbers k such that 9*10^k+1 is prime.

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A101397 Numbers k such that 4*10^k+3 is prime.

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A177506 Primes of the form 4*10^n+1.

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A101394 Numbers k such that 4*10^k+9 is prime.

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A101395 Numbers k such that 4*10^k+7 is prime.

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A101712 Indices of primes in sequence defined by A(0) = 41, 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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