A049054 -id:A049054 - cp's OEIS frontend

A159352 Primes of the form "1 [0]_n 3" - with zeros between 1 and 3.

13, 103, 100003, 1000003, 100000000003, 100000000000000003, 1000000000000000003, 1000000000000000000000000000000000000003, 100000000000000000000000000000000000000000000000000000003

Offset: 1

Parthasarathy Nambi, Apr 11 2009

			1000000000000000000000000000000000000003 is a prime with 38 zeros between 1 and 3.

Dario Alejandro Alpern, Factorization using the Elliptic Curve Method
Makoto Kamada, Prime numbers of the form 100...003.
Index entries for primes involving repunits.

Cf. A159031.

Cf. A049054 (numbers n such that 10^n + 3 is prime), A102006 (numbers n such that 10*10^n + 3 is prime), A011557 (powers of 10).

[p: n in [1..100] | IsPrime(p) where p is 10^n+3 ]; // Klaus Brockhaus, Apr 12 2009

select(isprime, [seq(10^k+3, k=1..998)]); # Robert Israel, Dec 28 2015

A102006 Indices of primes in sequence defined by A(0) = 13, A(n) = 10*A(n-1) - 27 for n > 0.

Original entry on oeis.org

0, 1, 4, 5, 10, 16, 17, 38, 55, 100, 104, 106, 122, 412, 425, 2606, 7667, 10469, 11020, 17752, 26926, 60775, 98287, 300475

Offset: 1

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

Numbers n such that 10*10^n + 3 is prime.
Numbers n such that digit 1 followed by n >= 0 occurrences of digit 0 followed by digit 3 is prime.
Numbers corresponding to terms <= 425 are certified primes.
No other terms <99,999.

			100003 is prime, hence 4 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...003.

Cf. A000533, A002275, A049054, A159352.

a=13;for(n=0,1500,if(isprime(a),print1(n,","));a=10*a-27)

for(n=0,1500,if(isprime(10*10^n+3),print1(n,",")))

a(n) = A049054(n) - 1.

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 02 2008
a(22)=60775, a(23)=98287 from Robert Price, Mar 03 2011
a(24) from A049054 by Ray Chandler, May 01 2015

A110980 Integers n such that 10^n+99 is prime.

Original entry on oeis.org

1, 2, 4, 6, 13, 14, 16, 17, 19, 30, 31, 60, 68, 73, 113, 144, 276, 288, 364, 449, 473, 739, 833, 1171, 1732, 2292, 3912, 7673, 9458, 16982, 19751, 21479, 23837, 77726

Offset: 1

Julien Peter Benney (jpbenney(AT)ftml.net), Sep 30 2005

The next term, if one exists, is >100000. - Robert Price, Apr 25 2011

See Kamada link - primecount.txt for terms, primesize.txt for discovery details including probable or proved primes - search on "10w99".

			If n=6, we have 10^6+99 = 1000000+99 = 1000099, 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.

[n: n in [1..400]| IsPrime(10^n+99)]; // Vincenzo Librandi, Nov 02 2014

Select[Range[78000],PrimeQ[10^#+99]&] (* Harvey P. Dale, Aug 23 2013 *)

a(1)=1 added by Vladimir Joseph Stephan Orlovsky, May 02 2008
a(29)-a(33) from Robert Price, Mar 22 2010
a(34)=77726 from Robert Price, Mar 03 2011

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

A110918 Integers n such that 10^n+91 is a prime number.

Original entry on oeis.org

1, 2, 3, 4, 11, 12, 15, 19, 136, 144, 732, 5754, 6602, 23499, 39583, 74254, 93356, 94016

Offset: 1

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

No additional terms < 100000.

See Kamada link - primecount.txt for terms, primesize.txt for discovery details including probable or proved primes - search on "10091".

			n = 4 is a member: 10^4+91 = 10000+91 = 10091, 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.

[n: n in [1..400]| IsPrime(10^n+91)]; // Vincenzo Librandi, Nov 02 2014

Select[Range[2000], PrimeQ[10^# + 91] &] (* Vincenzo Librandi, Nov 02 2014 *)

a(1)=1 added by Vladimir Joseph Stephan Orlovsky, May 02 2008
a(12)-a(15) from Robert Price, Dec 12 2010
Edited by Ray Chandler, Dec 23 2010
a(16)=74254 from Robert Price, Mar 03 2011
a(17)=92178 and a(18)=94016 from Robert Price, Apr 19 2011
a(17)=93356 corrected by Robert Price, Apr 19 2011
a(12) corrected by Tyler Busby, May 03 2024

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

A257636 Numbers n such that the base 10 reversals of n and n+1 are both prime.

Original entry on oeis.org

2, 13, 16, 30, 31, 34, 37, 70, 73, 91, 97, 106, 112, 118, 124, 130, 133, 145, 151, 166, 181, 199, 300, 310, 346, 358, 361, 364, 370, 376, 382, 388, 391, 700, 709, 721, 727, 730, 739, 745, 751, 754, 757, 760, 763, 775, 778, 784, 787, 790, 904, 907, 916, 919

Offset: 1

Robert Israel, Nov 04 2015

n such that n and n+1 are in A095179.
Leading 0's in the reversals are allowed.
Heuristically, the abundance of these numbers should be roughly similar to that of the twin primes.  Thus the sequence should be infinite but the sum of the reciprocals should converge.
All terms == 1 (mod 3) except for 2 and 3*10^k where k is in A049054.

			13 is in the sequence because both 31 and 41 are prime.

Robert Israel, Table of n, a(n) for n = 1..10000
Math StackExchange, Consecutive numbers where their revers numbers are primes

Cf. A000040, A004086, A049054, A095179.

revdigs:= proc(n) option remember; local x;
   x:= n mod 10;
   x*10^ilog10(n)+revdigs((n-x)/10);
end proc:
for i from 0 to 9 do revdigs(i):= i od:
Rprimes:= select(isprime@revdigs, [$1..10^4]):
Rprimes[select(t -> Rprimes[t+1]-Rprimes[t]=1, [$1..nops(Rprimes)-1])]; # Robert Israel, Nov 04 2015

SequencePosition[Table[If[PrimeQ[IntegerReverse[n]],1,0],{n,1000}],{1,1}][[;;,1]] (* Harvey P. Dale, Jan 07 2024 *)

for(n=1, 1e3, if(isprime(eval(concat(Vecrev(Str(n))))) && isprime(eval(concat(Vecrev(Str(n+1))))), print1(n, ", "))) \\ Altug Alkan, Nov 04 2015

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

A363353 Numbers k such that 10^k + 3 is a semiprime.

Original entry on oeis.org

3, 4, 7, 8, 10, 15, 16, 21, 27, 30, 37, 42, 43, 54, 66, 77, 96, 114, 130, 132, 155, 156, 168, 182, 213, 294

Offset: 1

Robert Israel, Aug 16 2023

a(27) >= 306. - Amiram Eldar, Aug 17 2023

			a(3) = 7 is a term because 10^7 + 3 = 13 * 769231 is a semiprime.

Cf. A001358, A049054, A363352.

select(t -> numtheory:-bigomega(10^t+3) = 2, [$1..70]);

a(18)-a(26) from Amiram Eldar, Aug 17 2023

cp's OEIS Frontend

A159352 Primes of the form "1 [0]_n 3" - with zeros between 1 and 3.

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A102006 Indices of primes in sequence defined by A(0) = 13, A(n) = 10*A(n-1) - 27 for n > 0.

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A110980 Integers n such that 10^n+99 is prime.

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Magma

Mathematica

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

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PARI

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A110918 Integers n such that 10^n+91 is a prime number.

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Magma

Mathematica

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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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