A023237 -id:A023237 - cp's OEIS frontend

A055781 Primes q of the form q = 10p + 1, where p is also prime.

31, 71, 131, 191, 311, 431, 971, 1031, 1091, 1511, 1571, 1811, 1931, 2111, 2411, 2711, 3371, 3491, 3671, 4091, 4211, 4391, 4871, 5231, 5471, 5711, 6011, 6131, 6311, 6911, 7331, 7691, 8111, 8231, 8291, 8831, 9371, 10091, 10211, 10331, 10391, 10631

Offset: 1

Labos Elemer, Jul 13 2000

nonn

Corresponding values of p in A023237. - Jaroslav Krizek, Jul 14 2010
From Sergey Pavlov, Jun 14 2017: (Start)
Let a, b, and c be prime numbers such that c = 10b + 1 = 10 * (10a + 1) + 1. Then c = 311, b = 31, a = 3. (There are no other solutions since any prime p > 3 is either of the form 3k + 1 or 3k - 1. In other words, while a > 3 and a, b are primes, a == 1 (mod 3), b == -1 (mod 3), whereas c == 0 (mod 3).)
So is for any similar sequence of primes (of the form kn + 1) where 2k + 1 == 0 (mod 3), e.g., for A002144: the equation of the form c = kb + 1 = k * (ka + 1) + 1 while a, b, c are primes could have the only solution iff a = 3 (but also could have not).
(End) [This comment needs to be rewritten. - N. J. A. Sloane, Feb 18 2019]

			1031 = 103*10 + 1, 1 appended to 103.

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

Cf. A005384, A005385, A023237.

select(isprime, map(t-> 10*t+1, select(isprime, [3,seq(i,i=7..2000,6)]))); # Robert Israel, Jun 13 2017

Select[10Prime[Range[200]]+1,PrimeQ]  (* Harvey P. Dale, Feb 04 2011 *)

is(n)=n%10==1 && isprime(n) && isprime(n\10) \\ Charles R Greathouse IV, Jun 17 2017

A075704 p and 12*p+1 are both primes.

Original entry on oeis.org

3, 5, 13, 19, 23, 29, 31, 59, 61, 71, 73, 83, 89, 101, 103, 139, 149, 191, 199, 223, 229, 233, 269, 271, 281, 293, 311, 379, 383, 401, 409, 433, 463, 479, 503, 523, 569, 601, 631, 643, 661, 691, 719, 751, 761, 773, 811, 829, 839, 863, 883, 929, 953, 1009, 1013

Offset: 0

Jani Melik, Oct 02 2002

nonn

			5 is a prime and 12*5+1=61 is also a prime. 13 and 12*13+1=157 are both primes...

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A005384, A023212, A007693, A023228, A023237.

ts_m_sophie_germain_pras := proc(n); if (isprime(n)='true' and isprime(12*n+1)='true') then RETURN(n); fi; end: seq(ts_m_sophie_germain_pras(i), i=1..2030);

Select[Prime[Range[300]],PrimeQ[12#+1]&] (* Harvey P. Dale, Feb 06 2012 *)

A105435 Primes which with a 1 prepended remain prime.

Original entry on oeis.org

3, 7, 13, 31, 37, 67, 73, 79, 97, 103, 109, 151, 163, 181, 193, 223, 229, 277, 283, 307, 367, 373, 409, 433, 439, 487, 499, 523, 571, 601, 607, 613, 619, 709, 733, 787, 811, 823, 877, 907, 997, 1069, 1087, 1093, 1117, 1171, 1213, 1279, 1321, 1399, 1423, 1447

Offset: 1

Lekraj Beedassy, Apr 08 2005

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A023237.

with(numtheory); for i from 1 to 500 do if isprime(10^(ceil(log[10](ithprime(i))))+ithprime(i)) then printf(`%d,`, ithprime(i)) fi: od: # James Sellers, Apr 09 2005

Select[ Prime[ Range[ 230]], PrimeQ[ FromDigits[ Join[{1}, IntegerDigits[ # ]]]] &] (* Robert G. Wilson v, Apr 09 2005 *)
Select[Prime[Range[250]],PrimeQ[10^IntegerLength[#]+#]&] (* Harvey P. Dale, May 30 2023 *)

Corrected and extended by Robert G. Wilson v and James Sellers, Apr 09 2005

A228857 Odd primes p > 3 for which 14*p+1 is also prime.

Original entry on oeis.org

5, 17, 47, 53, 59, 83, 107, 113, 149, 167, 173, 239, 269, 353, 419, 443, 449, 503, 509, 563, 587, 599, 647, 659, 677, 719, 797, 827, 929, 947, 977, 983, 1097, 1103, 1109, 1187, 1193, 1223, 1229, 1259, 1289, 1367, 1409, 1427, 1433, 1439, 1493, 1523, 1667

Offset: 1

Ant King, Sep 06 2013

nonn

In 1823, Legendre proved that the first case of Fermat’s Last Theorem is true for all exponents that are members of this sequence (see Ribenboim’s reference, p.112).

			As both 5 and 14*5 + 1 = 71 are prime, then 5 is a member of this sequence.

Paulo Ribenboim; Fermat’s Last Theorem For Amateurs, Springer-Verlag, New York, Inc., (1999).

Cf. A005384, A023212, A023228, A023237, A155943.

[p: p in PrimesInInterval(5,2000) |IsPrime(14*p+1)]; // Vincenzo Librandi, Sep 18 2016

Select[Prime[Range[3,1667]],PrimeQ[14#+1] &]

lista(nn) = forprime(p=5, nn, if(isprime(14*p+1), print1(p, ", "))); \\ Altug Alkan, Sep 18 2016

A243962 Primes p such that 10p + 1, 100p + 1 and 1000p + 1 are also primes.

Original entry on oeis.org

7, 13, 19, 103, 823, 1021, 1579, 1867, 2503, 3331, 5779, 6871, 6949, 9007, 10093, 10399, 11317, 11743, 13327, 13381, 15859, 16657, 17539, 17659, 22189, 26317, 26557, 26821, 27397, 27943, 29209, 29383, 30211, 32443, 37309, 38287, 40213, 40507, 44497, 47569, 47977

Offset: 1

K. D. Bajpai, Jun 16 2014

nonn

			7 is in the sequence because 7 is prime, 10*7 + 1 = 71 is prime, 100*7 + 1 = 701 is prime, and 1000*7 + 1 = 7001 is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A023237, A024912.

with(numtheory):A243962:= proc() local p; p:=ithprime(n); if isprime(10*p+1) and isprime(100*p+1) and isprime(1000*p+1) then RETURN (p); fi; end: seq(A243962 (), n=1..5000);

Select[Prime[Range[10000]], PrimeQ[10 # + 1] && PrimeQ[100 # + 1] && PrimeQ[1000 # + 1] &]

A326398 a(n) is the smallest k > 0 such that the concatenation prime(n)k is composite.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1

Offset: 1

David James Sycamore, Jun 07 2020

a(n) can only be 1 or 2 for any n >= 1. It appears that there are many more primes for which k = 1 than those for which k = 2 (~ 91% of the first 10^7 primes have k = 1).

			a(1) = 1 because 21 is prime, a(2) = 2 because 31 is prime (as 3 is in A023237), and 32 is composite

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A023237.

P := proc (n)
if isprime(10*ithprime(n)+1) then return 2 else 1;
end if:
end proc;
P(50);
seq(P(k), k = 1 .. 50);

Table[k = 1; p = Prime[n]; While[PrimeQ[10^IntegerLength[k]*p + k], k++]; k, {n, 120}] (* Michael De Vlieger, Jun 30 2025 *)

a(n) = my(k=1, p=Str(prime(n))); while (isprime(eval(concat(p, Str(k)))), k++); k; \\ Michel Marcus, Jun 07 2020

a(n) = 2 if prime(n) is in A023237, else a(n) = 1. [corrected by Chai Wah Wu, Jul 06 2020]

A356498 Primes p such that 100*p + 11 is also prime.

Original entry on oeis.org

2, 3, 23, 41, 83, 101, 107, 113, 137, 179, 233, 239, 251, 281, 293, 353, 359, 401, 419, 479, 503, 557, 563, 569, 587, 683, 701, 743, 809, 839, 857, 863, 941, 953, 977, 1049, 1091, 1103, 1193, 1217, 1277, 1283, 1361, 1367, 1427, 1487, 1499, 1523, 1607, 1619, 1847, 1871, 1877, 1889, 1907, 1949, 1973

Offset: 1

Daniel Blam, Aug 09 2022

nonn

100*p + 11 has the effect of appending 11 to p.

Primes of the form 3*k + 1 are never in this sequence, as 100*(3*k + 1) + 11 can be simplified to 3*(100*k + 37).

			2 is a term, as 100*2 + 11 is 211, which are both prime.
101 is a term, as 100*101 + 11 is 10111 which are both prime.

Cf. A000040, A002476 (primes of the form 3*k+1).

Similar to A023237.

from sympy import isprime
print([k for k in range(2000) if isprime(100*k+11) and isprime(k)])

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A055781 Primes q of the form q = 10p + 1, where p is also prime.

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A075704 p and 12*p+1 are both primes.

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Maple

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A105435 Primes which with a 1 prepended remain prime.

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Maple

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A228857 Odd primes p > 3 for which 14*p+1 is also prime.

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Magma

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A243962 Primes p such that 10p + 1, 100p + 1 and 1000p + 1 are also primes.

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Maple

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A326398 a(n) is the smallest k > 0 such that the concatenation prime(n)k is composite.

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Maple

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A356498 Primes p such that 100*p + 11 is also prime.

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