A088275 -id:A088275 - cp's OEIS frontend

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

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

A102008 Indices of primes in sequence defined by A(0) = 19, A(n) = 10*A(n-1) - 81 for n > 0.

Original entry on oeis.org

0, 1, 2, 3, 8, 17, 21, 44, 48, 55, 68, 145, 201, 271, 2729, 2840, 4561, 31809, 43185, 48108, 92690

Offset: 1

Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Dec 28 2004

Numbers n such that 10*10^n + 9 is prime.
Numbers n such that digit 1 followed by n >= 0 occurrences of digit 0 followed by digit 9 is prime.
Numbers corresponding to terms <= 271 are certified primes.
a(22) > 2*10^5. - Robert Price, Oct 11 2015

			1009 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 100...009.

Cf. A000533, A002275, A088275.

Select[Range[0, 200000], PrimeQ[10*10^# + 9] &] (* Robert Price, Oct 11 2015 *)

a=19;for(n=0,1500,if(isprime(a),print1(n,","));a=10*a-81)

for(n=0,1500,if(isprime(10*10^n + 9),print1(n,",")))

a(n) = A088275(n) - 1.

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

a(21) from Kamada data by Ray Chandler, May 01 2015

A107084 Integers k such that 10^k + 33 is prime.

Original entry on oeis.org

1, 3, 6, 9, 10, 31, 47, 70, 281, 366, 519, 532, 775, 1566, 1627, 2247, 2653, 4381, 4571, 7513, 10581, 13239, 15393, 72267, 105515, 215802

Offset: 1

Julien Peter Benney (jpbenney(AT)ftml.net), Jun 08 2005

The next term, if it exists, is > 39546. - Robert Price, Aug 21 2010
See Kamada link - primecount.txt for terms, primesize.txt for discovery details including probable or proved primes - search on "10033".
a(26) > 3*10^5. - Robert Price, Oct 26 2023

			For k = 3 we get 10^3 + 33 = 1000 + 33 = 1033, which is prime, so 3 is a term.

Makoto Kamada, Search for 10w33.
Index entries for primes involving repunits.

Cf. A049054, A088274, A088275, A095688, A108052, A108050, A108049, A108054.

a(20)-a(23) from Robert Price, Aug 21 2010
Edited by Ray Chandler, Dec 23 2010
a(24) from Robert Price, Jan 29 2011
a(26) from Robert Price, Oct 26 2023
a(25) from Kamada data by Tyler Busby, Apr 16 2024

A111021 Integers k such that 7*10^k + 31 is a prime number.

Original entry on oeis.org

1, 8, 11, 143, 203, 2727, 2911, 3339, 17039

Offset: 1

Julien Peter Benney (jpbenney(AT)ftml.net), Oct 04 2005

See Kamada link - primecount.txt for terms, primesize.txt for discovery details including probable or proved primes - search on "70031".

a(10) > 10^5. - Robert Price, Jan 28 2017

			k = 11 is a term because 7*10^11 + 31 = 7*100000000000 + 31 = 700000000000 + 31 = 700000000031, which is a prime.

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

Cf. A049054, A088274, A088275, A095688, A107083, A108049, A108050, A108052, A108054, A110980.

[n: n in [1..300] | IsPrime(7*10^n+31)]; // Vincenzo Librandi, Jul 03 2016

Select[Range[0, 10000], PrimeQ[7 10^# + 31] &] (* Vincenzo Librandi, Jul 03 2016 *)

a(9) from Ray Chandler, Dec 23 2010

a(1) = 1 prepended by Vincenzo Librandi, Jul 03 2016

A258932 Numbers k such that 10^k + 103 is prime.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 10, 11, 27, 35, 85, 169, 209, 221, 321, 347, 603, 610, 1229, 1391, 2171, 2303, 2679, 3977, 4545, 5721, 7090, 35877

Offset: 1

Vincenzo Librandi, Jun 15 2015

a(29) > 60000. - Michael S. Branicky, Apr 27 2025

			For n = 3, a(3) = 10^3 + 103 = 1103, which is prime.

Sequences of the type 10^n+k: A049054 (k=3), A088274 (k=7), A088275 (k=9), A095688 (k=13), A108052 (k=19), A108050 (k=21), A108312 (k=27), A107083 (k=31), A107084 (k=33), A135109 (k=37), A135108 (k=39), A108049 (k=43), A108054 (k=49), A135118 (k=51), A135119 (k=57), A135116 (k=61), A135115 (k=63), A135113 (k=67), A135114 (k=69), A135132 (k=73), A135131 (k=79), A137848 (k=81), A135117 (k=87), A110918 (k=91), A135112 (k=93), A135107 (k=97), A110980 (k=99), this sequence (k=103), A258933 (k=109), A165508 (k=111), A248349 (k=123456789), A248351 (k=987654321).

[n: n in [1..600] | IsPrime(10^n+103)];

Select[Range[5000], PrimeQ[10^# + 103] &]

is(n)=ispseudoprime(10^n+103) \\ Charles R Greathouse IV, Jun 13 2017

a(26)-a(28) from Jens Kruse Andersen, Jun 23 2015

A305531 Smallest k >= 1 such that (n-1)*n^k + 1 is prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 3, 10, 3, 1, 2, 1, 1, 4, 1, 29, 14, 1, 1, 14, 2, 1, 2, 4, 1, 2, 4, 5, 12, 2, 1, 2, 2, 9, 16, 1, 2, 80, 1, 2, 4, 2, 3, 16, 2, 2, 2, 1, 15, 960, 15, 1, 4, 3, 1, 14, 1, 6, 20, 1, 3, 946, 6, 1, 18, 10, 1, 4, 1, 5, 42, 4, 1, 828, 1, 1, 2, 1, 12, 2, 6, 4, 30, 3, 3022, 2, 1, 1

Offset: 2

Eric Chen, Jun 04 2018

nonn

a(prime(j)) + 1 = A087139(j).
a(123) > 10^5, a(342) > 10^5, see the Barnes link for the Sierpinski base-123 and base-342 problems.
a(251) > 73000, see A087139.

Eric Chen, Table of n, a(n) for n = 2..122
Gary Barnes, Sierpinski conjectures and proofs
Eric Chen, Table n, a(n) for n = 2..360 status

For the numbers k such that these forms are prime:
a1(b): numbers k such that (b-1)*b^k-1 is prime
a2(b): numbers k such that (b-1)*b^k+1 is prime
a3(b): numbers k such that (b+1)*b^k-1 is prime
a4(b): numbers k such that (b+1)*b^k+1 is prime (no such k exists when b == 1 (mod 3))
a5(b): numbers k such that b^k-(b-1) is prime
a6(b): numbers k such that b^k+(b-1) is prime
a7(b): numbers k such that b^k-(b+1) is prime
a8(b): numbers k such that b^k+(b+1) is prime (no such k exists when b == 1 (mod 3)).
Using "-------" if there is currently no OEIS sequence and "xxxxxxx" if no such k exists (this occurs only for a4(b) and a8(b) for b == 1 (mod 3)):
.
   b   a1(b)   a2(b)   a3(b)   a4(b)   a5(b)   a6(b)   a7(b)   a8(b)
--------------------------------------------------------------------
   2 A000043 ------- A002235 A002253 A000043 ------- A050414 A057732
   3 A003307 A003306 A005540 A005537 A014224 A051783 A058959 A058958
   4 A272057 ------- ------- xxxxxxx A059266 A089437 A217348 xxxxxxx
   5 A046865 A204322 A257790 A143279 A059613 A124621 A165701 A089142
   6 A079906 A247260 ------- ------- A059614 A145106 A217352 A217351
   7 A046866 A245241 ------- xxxxxxx A191469 A217130 A217131 xxxxxxx
   8 A268061 A269544 ------- ------- A217380 A217381 A217383 A217382
   9 A268356 A056799 ------- ------- A177093 A217385 A217493 A217492
  10 A056725 A056797 A111391 xxxxxxx A095714 A088275 A092767 xxxxxxx
  11 A046867 A057462 ------- ------- ------- ------- ------- -------
  12 A079907 A251259 ------- ------- ------- A137654 ------- -------
  13 A297348 ------- ------- xxxxxxx ------- ------- ------- xxxxxxx
  14 A273523 ------- ------- ------- ------- ------- ------- -------
  15 ------- ------- ------- ------- ------- ------- ------- -------
  16 ------- ------- ------- xxxxxxx ------- ------- ------- xxxxxxx
Cf. (smallest k such that these forms are prime) A122396 (a1(b)+1 for prime b), A087139 (a2(b)+1 for prime b), A113516 (a5(b)), A076845 (a6(b)), A178250 (a7(b)).

a(n)=for(k=1,2^16,if(ispseudoprime((n-1)*n^k+1),return(k)))

A110920 Integers n such that 2*10^n + 81 is a prime number.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 15, 36, 38, 51, 168, 1000, 2955, 8151, 16456, 17902, 18784, 24948, 28731, 87144

Offset: 1

Julien Peter Benney (jpbenney(AT)ftml.net), Oct 04 2005

See Kamada link - primecount.txt for terms, primesize.txt for discovery details including probable or proved primes - search on "20081".

a(22) > 10^5. - Robert Price, Jan 19 2017

			n = 3 is a term because 2*10^3 + 81 = 2*1000 + 81 = 2000 + 81 = 2081 and 2081 is prime.

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

Cf. A049054, A088274, A088275, A095688, A107083, A108049, A108050, A108052, A108054.

a(15)-a(18) from Kamada link by Ray Chandler, Dec 23 2010
a(1)-a(2) prepended by Robert Price, Jan 19 2017
a(19)-a(21) from Robert Price, Jan 19 2017

A110933 Integers k such that 3*10^k + 71 is a prime number.

Original entry on oeis.org

1, 4, 7, 16, 19, 190, 227, 235, 283, 319, 1655, 3955, 10666, 30724

Offset: 1

Julien Peter Benney (jpbenney(AT)ftml.net), Oct 04 2005

See Kamada link - primecount.txt for terms, primesize.txt for discovery details including probable or proved primes - search on "30071".

a(15) > 10^5. - Robert Price, Jan 22 2017

			k = 7 is a member because: 3*10^7 + 71 = 30000071, which is prime.

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

Cf. A049054, A088274, A088275, A095688, A107083, A108049, A108050, A108052, A108054.

isok(n) = isprime(3*10^n+71); \\ Michel Marcus, Jan 22 2017

a(13) from Ray Chandler, Dec 23 2010
Prepended a(1)=1 by Robert Price, Jan 22 2017
a(14) from Robert Price, Jan 22 2017

A110949 Integers n such that 4*10^n + 61 is prime.

Original entry on oeis.org

1, 2, 7, 11, 191, 248, 1067, 2666, 5252, 13400, 22886, 23739, 29095

Offset: 1

Julien Peter Benney (jpbenney(AT)ftml.net), Oct 04 2005

See Kamada link - primecount.txt for terms, primesize.txt for discovery details including probable or proved primes - search on "40061".

a(14) > 10^5. - Robert Price, Jan 22 2017

			n = 7 is in the sequence because 4*10^7 + 61 = 4*10000000 + 61 = 40000000 + 61 = 40000061, which is prime.

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

Cf. A049054, A088274, A088275, A095688, A107083, A108049, A108050, A108052, A108054.

isok(n) = isprime(4*10^n+61); \\ Michel Marcus, Jan 22 2017

a(9)-a(11) from Ray Chandler, Dec 23 2010
Prepended a(1)=1 by Robert Price, Jan 22 2017
a(12)-a(13) from Robert Price, Jan 22 2017

A110983 Integers k such that 5*10^k + 51 is prime.

Original entry on oeis.org

1, 3, 4, 16, 430, 727, 1415, 2691, 3160, 3904, 5464, 19875, 65255, 68524

Offset: 1

Julien Peter Benney (jpbenney(AT)ftml.net), Oct 04 2005

See Kamada link - primecount.txt for terms, primesize.txt for discovery details including probable or proved primes - search on "50051".

a(15) > 10^5. - Robert Price, Jan 28 2017

			k = 4 is a member because: 5*10^4+51 = 5*10000+51 = 50000+51 = 50051, which is prime.

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

Cf. A049054, A088274, A088275, A095688, A107083, A108049, A108050, A108052, A108054.

Select[Range[1, 1000], PrimeQ[5*10^# + 51] &] (* Julien Kluge, Dec 15 2016 *)

a(11)-a(12) from Ray Chandler, Dec 23 2010
Prepended a(1)=1 by Robert Price, Jan 28 2017
a(13)-a(14) from Robert Price, Jan 28 2017

cp's OEIS Frontend

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

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A102008 Indices of primes in sequence defined by A(0) = 19, A(n) = 10*A(n-1) - 81 for n > 0.

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A107084 Integers k such that 10^k + 33 is prime.

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A111021 Integers k such that 7*10^k + 31 is a prime number.

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Magma

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A258932 Numbers k such that 10^k + 103 is prime.

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Magma

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A305531 Smallest k >= 1 such that (n-1)*n^k + 1 is prime.

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PARI

A110920 Integers n such that 2*10^n + 81 is a prime number.

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